Faculty of technology and science
Department of Chemical Engineering
ARIF HUSSAIN
Evaluation of Different
Models for Flow through Paper
Chemical Engineering
Master thesis
Supervisor Lars Nilsson
Examiner Lars Jarnström
Karlstads universitet 651 88 Karlstad
Tfn 054-700 10 00 Fax 054-700 14 60
[email protected] www.kau.se
This thesis is dedicated to my parents and my
family for their love, endless support and
encouragement.
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Table of Contents
ABSTRACT.................................................................................................................................................. 4
SAMMANFATTNING PÅ SVENSKA ....................................................................................................... 4
1
SUMMARY .......................................................................................................................................... 4
2
INTRODUCTION ................................................................................................................................ 5
2.1
3
4
Flow Models. ................................................................................................................................ 8
2.1.1
Darcy‟s flow Equation. ......................................................................................................... 8
2.1.2
Forchheimer flow Equation. ................................................................................................. 9
2.1.3
Air Flow-Considering Compressibility Effects. ................................................................. 10
2.1.4
Missbach flow Model.......................................................................................................... 11
2.2
Flow Regimes in Porous Media .................................................................................................. 11
2.3
Aim of the present study. ............................................................................................................ 12
MATERIALS AND METHODS ........................................................................................................ 14
3.1
Laboratory Hand Sheet Former................................................................................................... 14
3.2
Bendtsen Air Permeance ............................................................................................................. 14
3.3
Modified Hand Sheet Former ..................................................................................................... 15
3.4
Air Flow Using Dewatering Equipment ..................................................................................... 15
3.5
Estimation of Pressure Increase. ................................................................................................. 15
3.6
Thickness Measurements ............................................................................................................ 16
3.7
Fiber Master Analysis ................................................................................................................. 16
3.8
Reproducability of Results. ......................................................................................................... 16
3.9
Air Leakage Experiments............................................................................................................ 16
RESULTS AND DISCUSSION ......................................................................................................... 17
4.1
Bendtsen Air Flow ...................................................................................................................... 17
4.2
Reynolds number for Bendtsen air flow. .................................................................................... 21
4.3
(Dewatering Equipment) Pressure Increase vs Dwell time. ........................................................ 21
4.4
Air Velocity vs. Sheet Basis Weight (Dewatering Equipment) .................................................. 22
4.5
Inertial effects and Turbulent flow.............................................................................................. 24
4.6
Flow Models for High Velocity Air Flow. ................................................................................. 24
4.7
Leakage Experiments .................................................................................................................. 26
5
CONCLUSION ................................................................................................................................... 27
6
NOMENCLATURE ........................................................................................................................... 28
7
REFERENCES ................................................................................................................................... 29
8
ACKNOWLEDGEMENT .................................................................................................................. 30
9
APPENDIX ......................................................................................................................................... 31
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ABSTRACT
For understanding the energy use during vacuum dewatering and through air drying
process, air flow through highly complex structure of paper has been investigated.
Experiments were performed for a wide range of pressure drops and basis weights. In
addition the pulp samples are refined to three different beating degrees. The calculated
Reynolds numbers, based on the fiber diameter, varies in a wide range between 0.0002 and
80. The majority of data at low Reynolds number (below approximately 0.2) agree rather
well with Darcy‟s law, so that the air flow is proportional to the pressure drop and inversely
proportional to the grammage. However, the data at high Reynolds number obtained from
air flow experiments using vacuum dewatering equipment, where large amount of air
sucked through the web have more news value. Different mathematical models for flow
through porous media are investigated to see how well they can describe the experimental
findings at high Reynolds numbers. It was found that for high Reynolds number flow, the
flow rate is a unique function of the quotient of pressure drop and grammage for a specific
degree of beating. It was also found that increased beating leads to a reduced air flow
through the sample. However, no clear conclusion regarding the importance of
compressibility and inertial forces when modeling the process could be made.
SAMMANFATTNING PÅ SVENSKA
För att bättre kunna beskriva energianvändning vid vakuumavvattning och
genomblåsningstorkning har luftflödet genom det komplexa pappersmaterialet mätts upp
för en lång rad olika tryckfall och ytvikter. Även malningens inverkan på luftflödet genom
pappersstrukturen har undersökts. Reynolds tal har bestämts baserat på fiberdiametern och
för de redovisade försöken varierar Reynolds tal i ett stort område mellan 0.0002 och 80.
Huvuddelen av de uppmätta flödeshastigheterna vid låga Reynolds tal (upp till ungefär 0.2)
stämmer relativt väl överens med Darcys lag, så att luftflödet är proportionellt mot
tryckfallet och omvänt proportionellt mot ytvikten. Vid högre värden på Reynolds tal har
luftflödesmätningar utförts i en utrustning konstruerad för att studera vakuumavvattning.
Vid vakuumavvattning strömmar stora luftmängder genom arket och de här data har större
nyhetsvärde. Olika matematiska modeller för strömning genom porösa material utvärderas
avseende hur väl de beskriver de uppmätta data vid höga värden på Reynolds tal. Det visar
sig att även vid höga värden på Reynolds tal är strömningshastigheten en unik funktion av
kvoten mellan tryckfall och ytvikt för en specifik malgrad. Ett annat resultat är att ökad
malning leder till ett minskat flöde genom arket. Det gick inte att dra några definitiva
slutsatser beträffande bidraget av friktion respektive tröghetskrafter till det totala tryckfallet
över arket.
1
SUMMARY
The flow rate of air through a porous material depends on the thickness, the pore size, pore shape and on
the viscosity of the flowing material and on the pressure drop. Usually the velocity (u), of a fluid flowing
through a porous material is modelled according to Darcy‟s law:
Equation 1
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where K represents the permeability of the material (influenced by pore size and shape); µ is the
viscosity, ΔP is the pressure drop and L is the thickness of the material.
Darcy‟s law is strictly applicable only for incompressible fluids. For higher flow rates, Darcy‟s law can
no longer be applied. The limit for applying Darcy‟s law is determined by the Reynolds number, and
often a limit for the applicability of Darcy‟s law is stated as Re < 1, although it is sometimes not so easy
to define the characteristic pore diameter in a stringent way. The purpose of the study is to give an
overview of flow rate of air through material in paper manufacturing, which is of great importance when
calculating the energy use for vacuum dewatering. Bendtsen air permeance tester and the vacuum
dewatering equipments were used. For measuring air flow through paper at lower pressure drops by
Bendtsen air permeance tester, sheets with different basis weights were prepared using laboratory hand
sheet former from unbeaten pulp and pulp beaten at three different numbers of revolutions. On the other
hand, for measuring air flow through paper at higher pressure drop or high air velocity by vacuum
dewatering equipment, sheets were prepared using modified hand sheet former typically designed for
dewatering equipment. Pressure drops of 0.74, 1.47 and 2.20 kPa were used to estimate air flow rate using
Bendtsen air permeance tester. Similarly for dewatering experiments, pressure drops of 20, 40 and 60 kPa
were used to estimate air flow through paper. Each laboratory sheets (20-300) g/m2 basis weight made
from unbeaten pulp and pup beaten at 1000, 2000 and 3000 revolutions was then investigated for air flow
at three different pressure drops of Bendtsen air permeance tester. For vacuum dewatering experiments,
the formed sheets were then placed into the sample holder of the vacuum dewatering apparatus. The
velocity was set so that dwell times of 1, 1.5, 2, 4, 8, 12, 16 and 20 ms were achieved. The plate is
accelerated rapidly and the air is able to pass through the sample at different dwell times and pressure
drop. The pressure difference in the vacuum tank was then recorded using Dewa-soft software by
exporting the files into MS-Excel. The sheets were then saved to measure the thickness later on. For
Bendtsen air permeance experiments, the applicability of Darcy‟s law was confirmed for sheets of
different basis weights (20-300) g/m2, produced from unbeaten pulp and pulp beaten at three different
revolutions. However, there are deviations from Darcy‟s law for low grammage sheets i.e. 20 g/m2. The
majority of the calculated Reynolds numbers are also in the range where Darcy‟s law is normally
expected to be applicable. Other results are that increased refining leads to less amount of air flow
through the sheets. Also, it was found that a lower grammage sheet accommodates large flow rates as
compared to higher grammage sheets.
However, the results from vacuum dewatering experiments show deviation from Darcy‟s law. The
calculated Reynolds numbers are high enough to follow laminar or viscous flow. The experimental results
are then compared with different theoretical models for flow through porous media. The model based on
theory in which inertial forces are dominating when taking compressibility into account seems to be
suitable for experiments at 2 ms dwell time. The deviations from Darcy‟s law are smaller for the 12 ms
dwell time. The reason behind this is not clear. However, the influence of any type of “offset” effects will
be greater when evaluating the flow rate based on the shorter dwell time 2 ms. It was found that for high
Reynolds number flow, the flow rate is a unique function of the quotient of pressure drop and grammage
for a specific degree of beating. Also the air flow, by decreasing the sheet basis weight increases at first
slightly, then with a big jump between 50 g/m2 and 20 g/m2 sheets basis weight. Further results are that,
the lower basis weight sheet allows more air to penetrate in terms of pressure increase through it. An
important finding was that superficial air velocity decreases as degree of beating increases.
2
INTRODUCTION
For predicting through air drying process and for the understanding of energy use during vacuum
dewatering in paper manufacturing, adequate description of the transport mechanisms involved is crucial.
Previously research made by different researchers is based on theory originally developed for flow
through porous media. This study dealt with the characterization of the paper structure and the description
of flow through porous media, especially its applicability to paper. Most of the theories presented by
different researchers were originally developed for well defined geometries of porous media, such as
packed spheres, and application to the highly complex structure of paper provides a new challenge.
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Paper is highly complex porous medium made up from wood which consists mainly of three different
polymeric substances: cellulose, hemicelluloses and lignin. Dullien (1979), define porous media as a
material or a structure which passes at least one of the following two tests.
1. “It must contain spaces, so called pores or voids, free of solids, imbedded in the solid or
semisolid matrix. The pores usually contain some fluid, such as air, water, oil, etc., or a
mixture of different fluids”.
2. “It must be permeable to a variety of fluids, i.e., fluids should be able to penetrate through
one face of a septum made of the material and emerge on the other side. In this case one
refers to a „permeable porous material”.
The structure of paper is strongly influenced by the raw material, i.e. tree species and the pulping process,
and also by the paper formation technique. Within the paper structure, fibers are oriented approximately
parallel to the sheet surface; in this plane however the fibers are more or less randomly oriented. The
process of dewatering and paper drying highly depends on the porous paper structure. The main
parameters describing the porous structure of paper are porosity, specific surface area, pore size
distribution and permeability. There are many different techniques for determining these parameters and it
is important to be aware that the obtained results depend on both the method used and on other factors
previously mentioned, such as pulp type and the paper formation process. The most important factor
determining the drying rate is the air permeability (Polat et al. 1992). “Permeability” is the term used for
the conductivity of the porous medium with respect to permeation by a Newtonian fluid. “Permeability”,
used in this general sense, is of limited usefulness only because its value in the same porous sample may
vary with the properties of the permeating fluid and the mechanism of permeation (Dullien, 1979).
Porosity is also an important parameter when discussing flow through porous medium. This is same as
volume fraction of gas in the material. When dealing with paper drying, where besides fibers and air also
water is present, it is often convenient to define the porosity or volume fraction of gas as follows:
Equation 2
Where εg, εs and εw are the volume fraction of gas, solid and water respectively, ρs is the fiber density, ρw
is the water density, G is the basis weight, R is the paper moisture ratio and z is the paper thickness. In
order to determine the porosity the fiber density, paper thickness, basis weight and moisture ratio have to
be determined. Measured porosities may vary slightly depending on method due to the compressibility of
paper and due to surface effects.
Another important parameter when discussing flow through porous medium is the specific surface area.
The specific surface area of paper is usually determined through optical methods (based on reflectance) or
solution or gas (N2) adsorption, in each case in conjunction with the BET method (Braunauer, Emmet and
Teller). By using some form of the Kozeny-Carman equation, which relates properties like porosity,
tortuosity and specific surface area to permeability, it is possible to determine the specific surface area
from gas and liquid permeation experiments. The average pore size and pore size distribution are also
important factors when it comes to flow through porous medium. There is no experimental technique to
directly describe the actual pore size (radius) of a paper structure since the shapes of the pores are highly
irregular and this parameter is not a single value but can be described only as an average pore size or by
the pore size distribution within the structure. A porous medium can be defined as a solid body which
contains void spaces or pores that are distributed randomly; without any conceivable pattern throughout
the structure of the solid body. Extremely small voids are called molecular interstices and very large ones
are called caverns or vugs. Pores (intergranular and intercrystalline) are intermediate between caverns and
molecular interstices. Fluid flow can only take place in the inter-connected pore space of the porous
media; this is called effective pore space.
Fluid flow through porous medium is an important subject. For flow through porous medium, it is
desirable to be able to predict the flow rate obtainable for a given energy input (usually measured as
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pressure drop) or to be able to predict the pressure drop necessary to achieve a specific flow rate. As the
drying rate in through-drying is very sensitive to the air flow rate, knowledge about the relationship
between air flow rate and applied pressure difference is critical for any prediction regarding dryer
capacity or economics. To describe the relationship between pressure drop and superficial velocity for
flow through porous media in case of paper material accurate models for describing this process are
lacking in the open literature. A few studies by previous researchers are; Polat et al. (1989), determined
for 25 and 50 g/m2 sheets basis weight and for superficial velocities in the range of 0.1 to 1.2 m/s. They
found the Missbach equation (equation 3) exponent (n) to range between 1.24 and 1.07.
Equation 3
Polat et al. (1992) measured the rate of through drying paper in the constant drying rate period for 210
combinations of temperature and through flow rate of air, basis weight, and initial moisture content of
paper. They found that for paper heavier than that which is through dried industrially, the Sherwood
number at high Reynolds numbers approaches independence from the paper thickness. However for thin
paper, they found that end effects predominate. Polat et al. (1993), in their experiments clearly
demonstrate that air flow through paper cannot be treated as purely viscous even at a flow rate of 0.6 m/s
for 150 g/m2 paper. They did experiments with through flow rates of 0.08-0.70 kg/m2s (0.07-0.60 m/s
superficial velocity) with air and with much higher kinematic viscosity, helium at through flow rates of
0.02-0.20 kg/m2s (0.12-1.20 m/s superficial velocity). The averaged measurements were then fitted to the
equation 3 to obtain the value of exponent n listed in Table 1.
The results presented in Table 1 also indicate that when air is replaced by helium, which has a kinematic
viscosity about 7.5 times higher than that of air, the inertial contribution to the pressure drop decreases. It
is however evident that even for helium flow there is substantial inertial contribution to the pressure drop
for the lowest basis weight i.e. 25 g/m2, since the value of the exponent n in equation 3 is higher than 1.0.
A value of n of 1.0 corresponds to a case without an inertial contribution.
Table 1 Exponent n for flow through dry paper of different basis weight sheets. (Polat et al. 1993)
Fluid
Air Flow
25
1.24
Basis weight, g/m2
50
100
150
1.07
1.05
1.02
250
1.01
1.08
1.01
1.00
1.00
1.00
Helium Flow
Polat et al. (1992-93), in their analysis of through air drying of paper have shown that basis weight,
moisture content, and furnish type can significantly affect the water removal rate during through air
drying. Weineisen et al. (2005) studied the effects of pore size distribution on the average drying rate.
They have shown how larger pores dry out and then act as bypass channels for the drying air. They
developed model in combination with correlations for heat and mass transfer for through drying of tissue
based on the equation of continuity. The model incorporates different geometric descriptions together
with corresponding heat and mass transfer correlations for flow through cylindrical conduits and flow
through packed beds of cylindrical fibers. At low air flow rates and low drying temperature, the model
shows very good agreement, however at higher drying intensities, the model predicts drying rates that are
generally higher than what is found in experimental studies described in the literature. Their results also
show that the early onset of the falling rate period in through drying at higher intensities could be
explained by channelling effects caused by a nonuniform pore size distribution. Weineisen et al. (2006),
present experimental data on through-drying of tissue sheets with three different basis weights i.e. 20, 30
and 40 g/m2 at intensities comparable to industrial conditions. Their superficial velocity was in the range 4
to 10 m/s and the corresponding pressure drop in the range 2 to 3.5 kPa. A power function was fitted to
the data and the exponent n of equation 3 was found to be 1.69, 1.59 and 1.56 for basis weight of 20, 30
and 40 g/m2 respectively. The higher values of the exponent n as compared to the work by Polat et al.
(1993) reflect a greater influence of inertial forces at the comparatively higher superficial velocities.
Weineisen et al. (2007) developed a model for through-drying of paper at high drying air temperature and
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constant pressure drop. Their model was solved for different combinations of the variables basis weight,
drying air temperature and pressure drop corresponding to industrial conditions and the results were
compared with data from bench-scale experiments. Their simulation show that the drying rate curve is
very sensitive to the air flow rate and that correctly modelling the correlation between pressure drop and
air flow rate is the most important factor. However, for a given basis weight, an increase in pressure drop
yielded fitted parameters that were somewhat different, i.e. a lower initial gas fraction and a higher
tortuosity, a change which increases the resistance to air flow. This means that in their model the
correlation between pressure drop and air flow rate does not quite capture the nonlinear relationship
shown by the experiments.
Ramaswamy (2003), describe the role of air flow in vacuum dewatering. He lists models based on physics
of the process primarily related to flow through porous media. First he shows the very basic equation
describing flow through a straight capillary of known radius i.e. well known Hagen-Poiseuille equation
for creeping flow through single straight capillary relating the frictional pressure drop to the liquid
velocity, and is given by
Equation 4
Then he describes the Darcy‟s law for flow through porous media, which is originally an extension of the
Hagen-Poiseuille equation for laminar flow through straight capillary. Darcy‟s law considers only the
fluid viscous effects neglecting the inertial effects, and is generally said to be valid for Reynolds numbers
less than unity. Commonly accepted form of Darcy‟s law is given as
Equation 5
Then he describes the fluid flow in which both viscous and inertial effects are taken into account.
Ramaswamy explain the findings by (Polat et. al. 1992-93), that the relationship between air flow rate and
pressure difference across the mat can be given by the Forchheimer relation as
Equation 6
Ramaswamy (2003) also describes one model for air flow during vacuum dewatering considering
compressibility effects. He explains that the applied pressure differences during the vacuum dewatering
can be quite high, of the order of 80 kPa; therefore one has to consider the compressibility effects of the
air as it passes through the sheet. For isothermal compressible flow, the modified form of Forchheimer
relation including viscous and inertial effects can be given as
Equation 7
Nilsson and Stenstrom, (1996) studied the permeability of pulp and paper. They modelled a sheet of paper
as a two dimensional network of cellulose fibers. They calculate the permeabilities and compared them
with measured values by solving stokes equation or equation for creeping flow through the structure.
They assumed structure to be highly ordered, when a fiber aspect ratio of 3.5 is used, and structure has
random distribution of fibers when a fiber aspect ratio of 5 is used.
2.1
Flow Models.
2.1.1 Darcy’s flow Equation.
Henri Darcy, a French civil engineer, in his 1856 publication laid the real foundation of the quantitative
theory of the flow of homogenous fluids through porous media. As a civil engineer, he was interested in
the flow characteristics of sand filters used to filter public water in the city of Dijon in France. The result
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of his classic experiments, globally known as Darcy‟s law, is thus stated: “The rate of flow V of water
through the filter bed is directly proportional to the area A of the sand and to the difference Δh in the
height between the fluid heads at the inlet and outlet of the bed, and inversely proportional to the
thickness L of the bed” and is given by,
Equation 8
Darcy‟s law represents a linear
relationship between the flow rate V and
the head (pressure gradient) Δh/L. Figure
1 shows how superficial velocity (mass
5
flow rate) depends on the pressure drop
4
according to Darcy‟s law. The constant
3
of proportionality K in the original Darcy
2
equation has been expressed as /k; is
the viscosity of the fluid and is called the
1
permeability of the porous medium.
0
Darcy‟s law is the most common way of
0
1
2
3
4
5
describing the laminar or viscous flow
Pressure drop
i.e. “linear relationship between the flow
rate and the pressure drop” through
porous media. According to Dullien Figure 1 Graphical illustration of relationship between Flow rate and
Pressure drop according to Darcy's law.
(1979);
Flow rate
Darcy
1. “Darcy‟s law assumes laminar or viscous flow (creep velocity); it does not involve the inertia
term (the fluid density). This implies that the inertia or acceleration forces in the fluid are being
neglected when compared to the classical Navier-Stokes equations”.
2. “Darcy‟s law assumes that in a porous medium a large surface area is exposed to fluid flow,
hence the viscous resistance will greatly exceed acceleration forces in the fluid unless turbulence
sets in”.
2.1.2 Forchheimer flow Equation.
Darcy‟s empirical flow model represents a simple linear relationship between flow rate and pressure drop
in porous media; any deviation from the Darcy flow scenario is termed non-Darcy flow. Physical causes
for these deviations are grouped under the following headings.
1.
2.
3.
4.
High velocity flow effects.
Molecular effects.
Ionic effects.
Non-Newtonian fluids phenomena.
However, in petroleum engineering, the most common phenomenon is the high flow rate effect. High
flow rate beyond the assumed laminar flow regime can occur in the following scenarios in petroleum
reservoirs. In 1901, Philippe Forchheimer, a Dutch man, while flowing gas thorough coal beds discovered
that the relationship between flow rate and potential gradient is nonlinear at sufficiently high velocity, and
that this non-linearity increases with flow rate. The additional pressure drop due to inertial losses is
primarily due to the acceleration and deceleration effects of the fluid as it travels through the tortuous
flow path of the porous media. The total pressure drop is thus given by Forchheimer empirical flow model
stated traditionally as;
Equation 9
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Incompressible flow
Superficial
velocity, (m/s)
The Forchheimer equation
assumes that Darcy‟s law is 1.2
still valid, but that an
Only frictional
additional term must be added
1
forces
to account for the increased
pressure drop. Hence this 0.8
Only intertial forces
equation will be called the
0.6
Darcy-Forchheimer
flow
model. Equation 9 is based on 0.4
fitting an empirical equation
50 % frictional
through experimental data. A 0.2
forces and 50 %
inertial forces
theoritical
model
was
0
constructed graphically, based
0
0.2 Pressure
0.4 drop, (kPa)
0.6
0.8
on the Forchheimer relation in
terms of incompressible flow Figure 2 Graphical illustration of relationship between Flow rate and Pressure
to compare it with the drop according to Forchheimer relation in terms of incompressible flow.
experimental results. For this purpose, maximum values of superficial velocity and the pressure drop were
assumed to be 1 m/s and 0.6 kPa respectively. The atmospheric pressure was assumed to be 1 bar. Then a
graph is constructed (Figure 2), which shows how the superficial velocity depends on pressure drop when
assuming only frictional forces, only inertial forces and 50% frictional & 50% inertial forces.
2.1.3 Air Flow-Considering Compressibility Effects.
Ramaswamy, S. (2003), explained that for air flow through porous media under lower pressure drop i.e.
as in the case of through air drying, the Forchheimer relation is directly applicable. He explained that the
applied pressure differences can be quite high during vacuum dewatering. Hence the compressibility
characteristics of the air as it passes through the sheet must be considered. He gives the modified form of
Forchheimer relation including viscous and inertial effects when considering compressibility
characteristics as.
Equation 10
A theoretical model was constructed based on the modified Forchheimer equation considering
compressibility effects. For this purpose, maximum values of superficial velocity and the pressure drop
were assumed to be 1 m/s and 0.6 kPa respectively. The atmospheric pressure was assumed to be 1 bar, so
that the assumptions are the same as those used for constructing Figure 2. Then a graph is constructed
(Figure 3), which shows how superficial velocity depends on the pressure drop when assuming only
frictional forces, only inertial forces and 50% frictional & 50% inertial forces.
Compressible flow
1.2
Superficial velocity, (m/s)
Only frictional
forces
1
0.8
Only intertial
forces
0.6
0.4
50 % frictional
forces and 50 %
inertial forces
0.2
0
0
0.2
Pressure
0.4 drop, (kPa)
0.6
0.8
Figure 3 Graphical illustration of relationship between Flow rate and Pressure
drop according to modified Forchheimer relation in terms of compressible flow.
10 | P a g e
2.1.4 Missbach flow Model.
In 1937 Missbach suggested that flow through porous media could be described by a power law model
(Weineisen et al 2006), and is given by:
Equation 11
Superficial velocity, (m/s)
Generally the exponent, n, in
equation 11 is in the range of 1 to
1.2
n = 1, Only frictional
2. For n=1, the equation reduces to
1
forces (Darcy's Law)
Darcy‟s law (equation 9) with a =
n = 2 (Inertial forces
0.8
/k. For n=2 the inertial losses
dominating)
dominate and may be interpreted as
0.6
n = 1.5 (frictional and
the constant βρ in equation 10. A
Inertial forces)
0.4
model was constructed graphically,
based on the Missbach flow
0.2
equation. For this purpose,
0
maximum values of superficial
0
0.2
0.4
0.6
0.8
velocity and the pressure drop were
Pressure Drop, (kPa)
assumed to be 1 m/s and 0.6 kPa
respectively as when constructing
Figure 2 and 3. Then a graph is
Figure 4 Graphical illustration of relationship between Flow rate and Pressure
constructed (Figure 4), which drop according to Missbach Flow Model.
shows how superficial velocity
depends on pressure drop when assuming only frictional forces i.e. n=1, only inertial forces i.e. n=2 and
50% frictional & 50% inertial forces i.e. n=1.5.
2.2
Flow Regimes in Porous Media
In fluid mechanics, the Reynolds number (Re) is a dimensionless number that gives a measure of the ratio
of inertial forces to viscous forces. Laminar flow occurs at low Reynolds number, where viscous forces
are dominant, and is characterized by smooth, constant fluid motion, while turbulent flow occurs at high
Reynolds numbers and is dominated by inertial forces. It was pointed out by Scheidegger (1960) that for
various porous media the value of the Reynolds number above which Darcy‟s law is no longer valid has
been found to range between 0.1 and 75. Change between laminar and turbulent flow is not distinct. There
is a transition which means that there is no accurate value for Reynolds number where a flow can be
considered to be laminar or turbulent. In general it can be said that transition to turbulent happens at
Reynolds number between 1000 and 10000. Since the transition from viscous to turbulent flow should
occur at an even higher Reynolds number the breakdown of Darcy‟s law must be due to some other
process than the transition from viscous to
turbulent flow. Polat et al. (1992) explain the
brake down of Darcy‟s law as being associated
with inertial effects occurring when the
streamlines of the flowing medium are distorted
due to changes in direction of motion big
enough for inertial forces to become significant
compared to viscous forces. The mechanisms of
losses due to inertial effects at relatively low
Reynolds numbers and due to turbulent losses
are identical. The mechanism behind inertial
Figure 5 Laminar vs. Turbulent Flow Regimes
effects in porous media differs from the effects
of turbulence only in that the change in direction of motion is induced by the structure of the porous
medium rather than being the result of directional change due to turbulent eddies. Because of this
mechanistic similarity, flow through paper in regions where inertial effects are important even though the
11 | P a g e
flow is not yet turbulent can be described by the same equations as those used for turbulent flow through
porous media.
Equation 12
In the literature, depending on the flow velocity and the nature of the porous media different flow patterns
have been observed. However four major regimes, using laser anemometry and visualization technique
proposed were.
These four regimes are;
a) “Darcy or laminar flow where the flow is dominated by viscous forces, here the pressure gradient
varies strictly linearly with the flow velocity. The Reynolds number at this point is less than 1”.
b) “At increasing Reynolds number, a transition zone is observed leading to flow dominated by inertia
effects. This begins in the range Re=1~10. This laminar inertia flow dominated region persists up to
and Re of ~150”.
c) “An unsteady laminar flow regime for Re =150 ~ 300 is characterized by occurrence of wake
oscillations and development of vortices in the flow profile”.
d) “A highly unsteady and chaotic flow regime for Re > 300, it resembles turbulent flow in pipes and is
dominated by eddies and high head losses”.
However there is large variation in the limiting Reynolds number for these transition zones as published
in the literature, therefore one cannot be too categorical about limits and transition zones as it relates to
the Reynolds number in porous media.
2.3
Aim of the present study.
Superficial velocity, (m/s)
This study presents experiments for evaluating the flow rate through paper material using two different
equipments. The Bendtsen air permeance tester which corresponds to low volumetric flow rates and
pressure drops. The other equipment should be vacuum dewatering which is relevant for industrial
conditions in terms of pressure drop over the sample and the flow rates. In paper manufacturing, the flow
rate of air through the material is of great importance when calculating the energy use for vacuum
dewatering. Darcy‟s law is only applicable to incompressible fluids and for low flow rates. The aim of
this study is to evaluate different
100
models and proposed some
suitable model for higher pressure
10
drops and higher flow rates
Polat et al. (1993)
through paper sheets of different
1
basis weights. The most relevant
0
20
40
60
80 Weineisen et al. (2006)
experimental data on through
drying has previously been
0.1
Pressure drop, (kPa)
published by other researchers,
Hussain (2011)
i.e.
(Polat et al 1993 &
0.01
Weineisen et al. 2006). However,
most of the previously published
0.001
data originate from experiments
performed at relatively low
0.0001
intensities, i.e. low air flow rates
and pressure drop. In present
study, pressure drops of 0.74,
1.47 and 2.20 kPa were used to Figure 6 Superficial velocity as a function of sheet basis weight, for comparison
estimate air flow rate through the between present study and relevant study did by previous researches.
paper of different basis weight
using Bendtsen equipment. The superficial air velocity was in the range of (0.00015-0.147) m/s. Similarly
12 | P a g e
for dewatering experiments, pressure drops of 20, 40 and 60 kPa were used to estimate air flow through
paper of different basis weight. The superficial air velocity was in the range of (1.64-68) m/s. Figure 6
and 7 shows the comparison between the estimated superficial velocities, working pressure drop and sheet
basis weight used in experiments of air flow through the paper material in this study with relevant studies
done by previous researchers.
Sheet basis weight
(g/m2)
350
300
250
200
150
100
50
0
1
2
3
4
5
6
7
8
9
10
Weieisen et al. (2006) 20 30 40
Polat et al. (1993)
25 50 100 150 250
Hussain (2011)
20 50 80 100 120 150 180 200 250 300
Figure 7 Paper with different basis weights, for comparison between present
study and previously research.
13 | P a g e
3
MATERIALS AND METHODS
The material used was Södra gold eucalyptus pulp. The pulp samples were divided into four test portions:
one for unbeaten and three for beaten grades. The four test portions, each corresponding to (30.0 ± 0.5) g
of oven dry pulp, were soaked in 1.5 liters of water for more than four hours according to standard
procedure, as the dry matter content of the pulp was > 60 %. The pulps were then disintegrated according
to standard ISO 5263-1:2004 for 30,000 revolutions, as the dry matter content was > 20%. Following
disintegration the pulp suspensions were drained in a Buchner funnel before being diluted to a total mass
of 300±5 grams, corresponding to a mass fraction of 10 % stock. Three of four thickened pulp samples
were then beaten in the PFI mill to three different degrees (1000, 2000 and 3000 revolutions, respectively)
according to standard procedure (ISO 5264-2:2002). Each test portion i.e. unbeaten and beaten pulp
samples were then diluted with 15 liters distilled water to get 0.2 % stock consistency. For measuring air
flow through paper at lower pressure drops by Bendtsen air permeance tester, sheets with different basis
weights were prepared using laboratory hand sheet former from unbeaten and three different beaten pulp
samples. On the other hand, for measuring air flow through paper at higher pressure drop or high air
velocity by vacuum dewatering equipment, sheets were prepared using modified hand sheet former
typically designed for dewatering equipment.
3.1
Laboratory Hand Sheet Former
Paper of basis weight 25, 50, 80, 100, 120, 150, 180, 200, 250 and 300 g/m2 from unbeaten and beaten
stock consistencies was prepared. The upper section of the laboratory hand sheet former is prefilled with
water to half its volume, followed by adding exact amount of stock to give desired basis weight sheet; to
ensure that fibres are not trapped in the wire during filling. The stock is filled up to the mark and
compressed air is used to agitate the system to ensure uniformity. After the agitation the drainage valve is
opened. The sheet formed on the wire is allowed to drain under reduced pressure for about 20 seconds.
Paper sheets of different basis weight from the laboratory hand sheet former were pressed between two
blotters and pressure is exerted on it by placing a couch on top of the blotters for 15 seconds after which
the couch is removed and the top blotter is removed and the sheet is allowed to stick to the lower blotter
from the wire. The complete stack of laboratory sheets was then wet pressed to 410 kPa for 25 seconds.
After the first pressing, a second pressing is carried out, for which the laboratory sheets were reversed and
all the blotters are replaced followed by raising the pressing pressure rapidly to 410 kPa for 2 minutes.
The laboratory sheets with different basis weight were then carefully separated from the blotters and are
mounted in a suitable manner in a conditioning room to prevent shrinkage. The sheets were then allowed
to dry for one day to be ready for air permeance testing.
3.2
Bendtsen Air Permeance
Laboratory sheets with different basis weight are tested
according to ISO standard (5636-3:1992E) method to determine
the air permeance at low pressure drops. Air permeance is
defined as “The mean flow of air through unit area under unit
pressure difference in unit time, under specified conditions and
at operating pressure” (ISO 5636-3:1992E) It is expressed in
micrometers per Pascal second [1 ml/ (m2Pas) = 1µm/ (Pas)].
The Bendtsen air permeance apparatus (Figure 8) consists of a
compressor, pressure controlling weights, sample clamping
device and the flow meter. The measuring head consists of a
device in which the test piece is clamped between an angular flat
surface and a circular rubber gasket. Each laboratory sheet (20300 g/m2 basis weight) made from unbeaten as well as beaten
was then clamped between an angular flat surface and a circular Figure 8 Bendtsen air Permeance Tester.
rubber gasket. The test area of each sheet was 10 cm2 ± 0.2 cm2. The air pressure was controlled by three
manostat weights provided with Bendtsen air permeance tester, which control the air pressure at 0.74
kPa±0.01 kPa, 1.47 kPa±0.02 kPa and 2.20 kPa±0.03 kPa. Air flow at three different pressure drops was
then recorded by pressing the handle above the measuring head to allow air to flow through the paper
14 | P a g e
sheets. For each sheet of different grammage air flow reading from flow meter was recorded. The sheets
were then saved to measure the thickness later on.
3.3
Modified Hand Sheet Former
Paper of basis weight 25, 50, 80, 100, 120, 150, 180, and 200 g/m2 from unbeaten and beaten stock
consistencies was prepared. A hand sheet mould with a rectangular top was slightly modified to be able to
produce circular sheets of 184 mm diameter. The modified hand sheet mould differed in some aspects
from the standard hand sheet mould in SCAN-CM 26:99. The hand sheet mould was filled up with water
to a total volume of 3.5 litres in the upper section. The exact amount of stock to give the desired basis
weight was added which results in a thin stock concentration of about 0.01 %. The stock was stirred by
moving a mechanical device up and down five times before the water was evacuated. The sheets were
formed by pulling the handle upwards to drain off the water.
3.4
Air Flow Using Dewatering Equipment
A bench-scale, laboratory vacuum dewatering was constructed as shown in Figure 9. Vacuum level, dwell
time and frequency of single vacuum pulses are in the range
typical for a Through Air Drying (TAD) tissue machine. The
vacuum dewatering equipment consists of two linear drives,
servo motor, and plate with slot width of 5 mm. The forming
fabric along with the formed sheet was taken from the
modified hand sheet former and placed into the sample
holder of the vacuum dewatering apparatus. The clamping
device was attached to the sample by two clamps. The
velocity was set so that dwell times of 1, 1.5, 2, 4, 8, 12, 16
and 20 ms were achieved and the start button was pressed for
each test to move the plate for the purpose of dewatering. For
each sheet sample having different basis weights prepared
from unbeaten and beaten pulp samples, the vacuum pressure
Figure 9 Vacuum Dewatering Apparatus.
was maintained at 20, 40 and 60 kPa in a vacuum tank using
a vacuum pump. When the plate is accelerated rapidly and
the slot passes under the sample holder, air is able to pass through the sample at different dwell times and
pressure drops resulting in sheet dewatering. The pressure difference in the vacuum tank was then
recorded using Dewa-soft software by exporting the files into MS-Excel.
3.5
Estimation of Pressure Increase.
Pressure increase/ difference from pressure
data obtained by Dewa-soft software are
something very difficult to evaluate. However
the approach used to estimate pressure
difference after exporting the pressure data
from Dewa-soft software into MS-Excel is
discussed below. By exporting pressure data
into MS-Excel following graph as shown in
Figure 10 in which pressure data as a function
of time is plotted was obtained. When the slot
of the plate passes over the vacuum tank a
vacuum pulse is created; air passes through the
sheet and the pressure in the vacuum tank
increases as shown in Figure 10. The set
pressure drop i.e. 20 kPa in Figure 10 was
named as P1, and the point where vacuum Figure 10 Graph obtained after exporting data from dewatering
pulse ends was named as P2. The pressure equipment into MS-Excel.
15 | P a g e
difference ΔP, was then calculated as ΔP = P2 – P1.
The sheets are then carefully detached from the forming fabric and are allowed to dry in a conditioning
room for thickness measurements. The mass flow rate (kg/sec) and superficial velocity (m/sec) was then
calculated from estimated pressure difference for each sheet sample using equation 13 and 14
respectively.
Equation 13
Equation 14
3.6
Thickness Measurements
Sheets with different basis weights, produced from unbeaten and beaten pulp samples are then tested to
measure thickness according to standard (ISO 534:1998); using thickness tester TJT-Teknik
manufactured by Lorentzen and Wettres AB. Sheet samples with different basis weight prepared from
laboratory hand sheet former and modified hand sheet former were then scanned between two opposite
measurement points so that a profile of the paper was obtained which gives a peak value corresponding to
the thickness of the sheet.
3.7
Fiber Master Analysis
The pulp samples unbeaten and beaten at three different beating
revolutions i.e. 1000, 2000 and 3000 revolutions are then collected for the
fiber analysis. L&W Fiber Tester measures bra length, width, fines, shape
factor and coarseness. The instrument has a sample feeder with six
positions designed as a rotating disc (Figure 11). This automates the
measurement procedure, making it easier for the operator. The total
measurement cycle is made within six minutes. One feature is that images
of fibers are displayed during measurements. It is also possible to save
images of detected objects for later viewing.
3.8
Reproducibility of Results.
Figure 11 Fiber Master Tester.
There is some spread in the data obtained during dewatering experiment; therefore some attention was
given to the reproducibility of these results. For this purpose a sheet of 20 g/m2 was prepared using
modified hand sheet former. After that sheet was carefully transfer to sample holder of vacuum
dewatering equipment. Parameters are set as 2.5 m/s plate velocity and 20 kPa pressure drop. The plate is
accelerated for allowing air to pass through the sheet. The same sheet is then tested for air flow up to five
times to see the difference in pressure increase. Another experiment was performed by making 50 g/m2
sheets using modified hand sheet former. All sheets were carefully transferred to sample holder of
dewatering equipment. Parameters are adjusted as 0.41 m/s velocity and 20 kPa pressure drop. Different
50 g/m2 sheets prepared exactly in the same way are tested for air flow up to four times. Finally three
sheets of 20 g/m2 were prepared. Parameters are set as 2.5m/s plate velocity and 20, 40 and 60 kPa
pressure drops for three consecutive sheets respectively. The plate is then accelerated for allowing air to
pass through the sheets at 20, 40 and 60 kPa. Sheets of grammage 50 g/m2 were then allowed to dry in an
oven at 105°C for basis weight estimation later on.
3.9
Air Leakage Experiments.
Finally some attention was given to investigate, if there is any leakage of air from the walls of sample
holder; when determining the air flow through paper. For this purpose, a rubber pad which does not allow
air to penetrate through it; was transfer to the sample holder of the vacuum dewatering equipment. The
plate velocities were set as 0.25, 0.41 and 2.5 m/s. For each velocity, plate is then accelerated up to four
times for allowing air to pass around the rubber pad at 20, 40 and 60 kPa pressure drops.
16 | P a g e
4
4.1
RESULTS AND DISCUSSION
Bendtsen Air Flow
A quantitative description of flow through porous media begins with an experimental rule found by
Darcy. This rule, describing that the rate of flow passing through a powder bed is proportional to the
pressure gradient of the flow, has been extended to a variety of porous media. There are various equations
to express this rule and its modification, known collectively as Darcy‟s law (equation 1).
The majority of the experimental results obtained for sheets of different basis weights (20-300) g/m2,
produced from unbeaten and beaten pulps using Bendtsen air permeance tester confirmed the applicability
of Darcy‟s law. Figures 12, 13, 14 and 15 shows the volumetric air flow rate, (ml/min) as a function of
pressure difference divided by sheet grammage (ΔP/G), for unbeaten pulp and pulp beaten at 1000, 2000
and 3000 beating revolutions respectively.
Volumetric air flowrate, (ml/min)
6000
5000
4000
0.74KPa
3000
1.47KPa
2000
2.20KPa
1000
0
0
0.01
0.02
0.03
0.04
0.05
ΔP/G, (m2kPa/g)
Figure 12 Volumetric air flow rate as a function of pressure difference/ grammage, for unbeaten pulp.
Volumetric air flowrate, (ml/min)
3000
2500
2000
0.74KPa
1500
1.47KPa
1000
2.20KPa
500
0
0
0.01
0.02
ΔP/G,
0.03
0.04
0.05
(m2kPa/g)
Figure 13 Volumetric air flow rate as a function of pressure difference/ grammage, for pulp beaten at 1000 revolutions.
17 | P a g e
Volumetric air flowrate, (ml/min)
1200
1000
800
0.74KPa
600
1.47KPa
400
2.20KPa
200
0
0
0.01
0.02
0.03
0.04
0.05
ΔP/G, (m2kPa/g)
Figure 14 Volumetric air flow rate as a function of pressure difference/ grammage, for pulp beaten at 2000 revolutions.
Volumetric air flowrate, (ml/min)
120
100
80
0.74KPa
60
1.47KPa
40
2.20KPa
20
0
0
0.01
0.02
0.03
0.04
0.05
ΔP/G, (m2kPa/g)
Figure 15 Volumetric air flow rate as a function of pressure difference/ grammage, for pulp beaten at 3000 revolutions.
It can be seen in Figures 12, 13, 14 and 15 that there is a linear relationship between the volumetric air
flow and the ΔP/G; which confirms the applicability of Darcy‟s law. One important finding during the
estimation of air flow through paper of different basis weights using Bendtsen equipment was that; by
increasing the degree of beating less amount of air flow through the sheets. The possible explanation is
the creation of fines during beating. Refinement and other mechanical treatment have been shown to
increase the specific surface area of fibers, as well as increase their swelling capacity, flexibility and
compressibility (Ramaswamy 2003). Although previous research on refining describes how it affects
sheet solid contents, how difficult vacuum dewatering will be then, but these entire findings correlate very
well to air flow through paper. Nordman, L. (1954) showed that increasing the content of fines decreases
the solids content obtainable by suction in a linear fashion. Britt and Unbehend (1985), on the other hand,
18 | P a g e
show that increasing the fines initially increases the level of dryness after vacuum dewatering but further
increase in the amount of fines makes vacuum dewatering more difficult.
Refining of pulp also improves the bonding ability of fibers, causing a variety of simultaneous changes in
fibers such as; internal fibrillation, external fibrillation, fiber shortening or cutting. Internal fibrillation of
fibers improves the flexibility and collapsibility of fibers, which further improves inter-fiber bonding.
Another explanation of low air flow by increasing degree of refining is that, refining makes the paper
more dense i.e. reduces the pore space between the fibers. The results from Fiber master analysis
however, show that there is slightly increase in fines by increasing degree of beating. Table 2 below
shows the results obtained from fiber master analysis for unbeaten and pulp beaten at three different
beating revolutions.
Table 2 Fiber master results for unbeaten and beaten pulp at three different revolutions.
Variable
Length
Width
Shape
Fines
Coarseness
No. Fibers in sample
Unbeaten
0,714 mm
16,8 μm
90.44%
4.80%
59,0 μg/m
2950558
Beaten 1000
0,726 mm
17,1 μm
91.00%
4.80%
59,5 μg/m
2886976
Beaten 2000
0,723 mm
17,3 μm
91.17%
5.10%
63,5 μg/m
2737249
Beaten 3000
0,716 mm
17,5 μm
91.29%
5.50%
69,4 μg/m
2578134
The flow rate of air through a porous material depends on the porosity. Results obtained from thickness
measurements also confirm that thickness and thereby porosity decreases as degree of beating increases.
Figure 16 shows the results obtained from thickness measurements for paper sheets produced from
unbeaten and beaten pulp.
600
500
Thickness, (µm)
400
UnBeaten
300
B1000
B2000
200
B3000
100
0
0
100
200
Basis Weight,
300
400
(g/m2)
Figure 16 Thickness as a function of sheet basis weight, measured for sheets made from laboratory hand sheet former for
Bendtsen experiments.
19 | P a g e
It can also be seen in Figure 16 that thickness increases as the sheet basis weight increases. As mentioned
above that flow rate of air though a porous material depends on the thickness, due to this low air flow
through higher grammage sheets are found.
8000
6000
0.74KPa
4000
1.47KPa
2000
2.20KPa
0
0
0.05
ΔP/G,
0.1
Volumetric air flow,
(ml/min)
8000
6000
0.74KPa
4000
1.47KPa
2000
2.20KPa
0
0.15
0
8000
6000
0.74KPa
4000
1.47KPa
2000
2.20KPa
0
0.05
0.1
0.05
ΔP/G,
10000
ΔP/G,
10000
(m2kPa/g)
Figure 17 Volumetric air flow rate as a function of
pressure difference/ grammage, for low grammage
sheets of unbeaten pulp
0
Volumetric air flow,
(ml/min)
10000
0.15
(m2kPa/g)
Figure 18 Volumetric air flow rate as a function of
pressure difference/ grammage, for low grammage
sheets of 2000 rev beaten pulp
0.1
0.15
(m2kPa/g)
Figure 19 Volumetric air flow rate as a function of
pressure difference/ grammage, for low grammage
sheets of 1000 rev beaten pulp
Volumetric air flow,
(ml/min)
Volumetrica air flow,
(ml/min)
The data obtained using Bendtsen air permeance show deviations from Darcy‟s law for low grammage
sheets i.e. 20 g/m2 for unbeaten and beaten pulp samples Figures (17, 18, 19 and 20). The possible reason
might be that the maximum range of measuring air flow using Bendtsen air permeance tester was 8820
ml/min, or for lower grammage sheets pin hole counts increases exponentially which accommodates large
flow rates. This is what Polat et. al. (1993), also observed in their experiments, they found that air flow
through paper cannot be treated as purely viscous even at a flow rate of 0.6 m/s for 150 g/m2 paper. They
did experiments with through flow rates of 0.08-0/70 kg/m2s (0.07-0.60 m/s superficial velocity) with air
and with much higher kinematic viscosity, helium at through flow rates of 0.02-0.20 kg/m2s (0.12-1.20
m/s superficial velocity). The results presented in Table 1 also indicate that when air is replaced by
helium, which has a kinematic viscosity about 7.5 times higher than that of air, the inertial contribution to
the pressure drop decreases. It is however evident that even for helium flow there is substantial inertial
contribution to the pressure drop for the lowest basis weight (25 g/m2). Weineisen at al. (2006), also in
their experiments found the value of exponent n in equation 3 for sheets of basis weight 20, 30 and 40
g/m2 in the range of 1.56 and 1.69. This indicates that, for flow through paper at high air velocities, the
pressure drop due to inertial effects cannot be neglected.
8000
6000
4000
0.74KPa
2000
1.47KPa
2.20KPa
0
0
0.05
0.1
0.15
ΔP/G, (m2kPa/g)
Figure 20 Volumetric air flow rate as a function of
pressure difference/ grammage, for low grammage
sheets of 3000 rev beaten pulp
20 | P a g e
4.2
Reynolds number for Bendtsen air flow.
Equation 15
Reynold's number, (Re)
In fluid mechanics, the Reynolds number (Re) is a dimensionless number that gives a measure of the ratio
of inertial forces to viscous forces. Laminar flow occurs at low Reynolds number, where viscous forces
are dominant, and is characterized by smooth,
0.18
constant fluid motion, while turbulent flow
0.16
occurs at high Reynolds numbers and is
dominated by inertial forces, which tends to
0.14
produce eddies, vortices and other flow
0.12
instabilities.
0.1
0.08
0.06
0.04
0.02
0
4.3
(Dewatering Equipment) Pressure
Increase vs. Dwell time.
Pressure Increase, (kPa)
The Reynolds number at which Darcy‟s law is
no longer applicable ranges between 0.1-75
depending on the porous structure and the
0
100
200
300
400
choice of characteristic dimension used (Polat et
2)
Sheet
basis
weight,
(g/m
al. 1992). The calculated Reynolds numbers for
unbeaten and beaten pulp to three revolutions Figure 21 Calculated Reynolds numbers as a function of sheet
using Bendtsen air flow experiments are very basis weight for Bendtsen air permeance experiments.
low, even the higher value is near to 0.18. Figure
21 shows the calculated Reynolds numbers as a function of sheet basis weights.
8.00
7.00
6.00
5.00
4.00
3.00
2.00
1.00
0.00
y = 0.3551x + 0.4072
Pressure Increase, (kPa)
R² = 0.9934
In order to investigate the relationship between
the air flow in terms of pressure increase and the
vacuum dwell time, eight sheets of 20 g/m2 were
prepared. The velocity was set so that dwell
times of 1, 1.5, 2, 4, 8, 12, 16 and 20 ms were
achieved and the start button was pressed for
each test sheet to move the plate at above
mentioned dwell times for the purpose of
0
5
10
15
20
25
dewatering. For each sheet sample, the vacuum
pressure was maintained at 60 kPa in a vacuum
Vacuum dwell time, (ms)
tank using a vacuum pump. Figure 22 shows the
way in which the air that penetrated the sample
as given by the pressure increase in the vacuum Figure 22 Pressure increase as a function of vacuum dwell time.
tank increases when the dwell time is increased.
7
The results showed that when the dwell time
6
was increased, the amount of air flowing
through the sample (in terms of pressure
5
20g/m2
increase in the vacuum tank) was also increased:
4
80g/m2
there is simply more time available for the air to
3
be sucked through the slots in the plate. The
120g/m2
2
increase in the air flow through the sheet that
180g/m2
1
resulted from increasing the dwell time was
0
observed for the unbeaten pulp samples. Figure
250g/m2
0
10
20
22 shows that there is a linear relationship
between the pressure increases as a function of
Vacuum dwell time, (ms)
vacuum dwell time. This leads to the conclusion
that the flow rate through the sample can be
calculated from one dwell time only. For the Figure 23 Pressure increase as a function of vacuum dwell time
for sheets of different basis weights.
21 | P a g e
majority of flow rates presented, a dwell time of 12 ms was used. However, a shorter dwell time of 2 ms
was also used.
Figure 23 shows the same relationship between the pressure increases as a function of vacuum dwell time
for sheets of different basis weight at 60 kPa. It can be seen a lower basis weight sheet allows more air to
penetrate in terms of pressure increase through it. The possible explanation is that, as the sheet basis
weight increases the thickness of the sheet increases; which means that by increasing paper thickness,
mean pore size decreases and the pore size distribution becomes narrower.
4.4
Air Velocity vs. Sheet Basis Weight (Dewatering Equipment)
Thickness, (μm)
The results from the dewatering experiments show that, air flow decreases as the sheet basis weight
increases. Also the air flow, by decreasing the sheet basis weight increases at first slightly, then with a big
jump between 50 g/m2 and 20 g/m2 sheets basis weight Figure 25, 26 and 27. The possible reason is that
as the sheet basis weight increases the thickness of sheet increases. In accord with Darcy‟s law, fluid flow
through porous media depends on the thickness of the porous material. Polat et al., (1993) also confirm in
their experiments on sheets basis weight between 25-250 g/m2 that “With decreasing basis weight the
permeability increases, at first slightly, then with a big jump between 50 g/m2 and 20 g/m2 paper”.
According to Polat et al. (1993) “the small increase in air flow with decreasing basis weight from 250 to
50 g/m2 probably derives from small changes in pore size and pore size distribution”. They also quote the
finding of Bliesner that “as paper thickness increases, mean pore size decreases and the pore size
distribution becomes narrower”. Another explanation of this is given by (Corte & Kallmes) and (Corte &
Lloyd) as quoted in article by Polat et al. that “For commercial paper, pin hole counts increases
exponentially with decreasing basis weight. Pin holes accommodate disproportionately large flow rates.
In other words, at high basis weight three-dimensional pores control the flow but, as basis weight
decreases,
Z-directional
pores
600
become increasingly important. The
large increase in permeability at
500
such a low basis weight thus reflects
a substantial fraction of flow
400
through Z-directional pores”. Less
UnBeaten
amount of air flow through higher
300
grammage sheets was also found
B1000
during estimation of air flow at low
200
B2000
pressure drops using Bendtsen air
permeance tester.
B3000
100
Figure 24 shows the thickness
values obtained for different
0
grammage sheets made from
0
50
100
150
200
250
unbeaten and beaten pulp samples. It
2
Sheet basis Weight, (g/m )
can be seen that thickness increases
as the sheet basis weight increases. Figure 24 Thickness measurements of sheets made from modified hand
However, the thickness values sheet former for dewatering experiments obtained using thickness testerobtained from sheets prepared with TJT-Teknik.
laboratory hand sheet former are
lower as compared to modified sheet former due to pressing effects.
The mass flow rate and the superficial air velocity was calculated at dwell time of 12 ms and at three
different pressure drops of 20, 40 and 60 kPa according to equation 16 and 17 respectively.
Equation 16
22 | P a g e
Equation 17
18.0
16.0
14.0
12.0
10.0
8.0
6.0
4.0
2.0
0.0
Superficial Air Velocity, (m/s)
Superficial Air Velocity, (m/s)
Figures 25, 26 and 27 below show the superficial air velocity as a function of sheet basis weight, for
dwell time of 12 ms. It can also be seen that superficial air velocity decreases as degree of beating
increases. This again confirms the theory of refining, which was observed in the case of Bendtsen
experiments. Refinement and other mechanical treatment have been shown to increase the specific surface
area of fibers. Ramaswamy (2003) describe that “Potential mechanisms of water removal by air flow
during vacuum dewatering include viscous drag by flowing air on the water present in the fiber interstices
or pores and compression by surface tension forces. Due to the level of fines, paper and board grades
offer significant resistance to air flow thus minimizing the amount of air flow under vacuum dewatering
conditions”. Although previous research on refining describes how it affects sheet solid contents, how
difficult vacuum dewatering will be then, but these entire findings correlate very well to air flow through
paper. The results from Fiber master analysis also confirm that an increase beating decreases the
permeance as fines are created. Table 2 above shows the results obtained from fiber master analysis for
unbeaten and pulp beaten at three different beating revolutions.
Unbeaten
1000 deg
2000deg
3000 deg
0
100
200
40.0
35.0
30.0
25.0
1000 deg
15.0
10.0
2000deg
5.0
3000 deg
0.0
300
Sheet basis weight,
Unbeaten
20.0
0
(g/m2)
100
200
300
Sheet basis weight, (g/m2)
Figure 25 Superficial air velocity (m/s) as a function of
sheet basis weight (g/m2), estimated at 20kPa and at 12
ms dwell time.
Figure 26 Superficial air velocity (m/s) as a function of
sheet basis weight (g/m2), estimated at 40 kPa and at 12
ms dwell time.
Superficial Air Velocity, (m/s)
70.0
60.0
50.0
40.0
Unbeaten
30.0
1000 deg
20.0
2000deg
10.0
3000 deg
0.0
0
100
200
Sheet basis weight,
300
(g/m2)
Figure 27 Superficial air velocity (m/s) as a function of sheet basis weight (g/m2), estimated at 60 kPa and at 12 ms dwell
time.
23 | P a g e
4.5
Inertial effects and Turbulent flow.
Reynolds numbers, (Re)
The application of Darcy‟s law has been the predominant way of describing air flow through paper, i.e.
flow through paper has generally been considered as being purely viscous. However, later research has
shown that when pinholes become evident the flow is no longer purely viscous and thus Darcy‟s law does
no longer apply. According to Polat et al. (1989 and 1992), since the findings that flow through thin paper
is not always purely viscous, many researchers have instead of treating this fact, made sure to perform
their experiments at viscous flow, i.e. by increasing paper thickness or decreasing flow rate. The
Reynolds number at which Darcy‟s law are no longer applicable ranges between 0.1-75 depending on the
porous structure and the choice of characteristic dimension used Polat et al. (1992). The calculated
Reynolds numbers at 2 and 12 ms dwell time for unbeaten and beaten pulps using dewatering equipment
experiments are high. Figure 28 shows the calculated Reynolds numbers as a function of sheet basis
weights for dewatering experiments performed at 2 and 12 ms dwell time. Polat et al. (1992) explain the
breakdown of Darcy‟s law being associated with inertial effects occurring when the streamlines of the
flowing medium are distorted due to changes in direction of motion big enough for inertial forces to
become significant compared to viscous forces. The mechanisms of losses due to inertial effects at
relatively low Reynolds numbers and
90
due to turbulent losses are identical. The
mechanism behind inertial effects in
80
porous media differs from the effects of
70
turbulence only in that the change in
direction of motion is induced by the
60
structure of the porous medium rather
50
than being the result of directional
20 kPa
change due to turbulent eddies. Because
40
40 kPa
of this mechanistic similarity, flow
30
through paper in regions where inertial
60 kPa
20
effects are important but the flow is not
yet turbulent can be described by the
10
same equations as those used for
0
turbulent flow through porous media.
0
100
200
Thus the Forchheimer relation (equation
2
9) and the equation of Missbach
Sheet basis weight, (g/m )
(equation 11) are both on principal
applicable to flow through paper. The
Figure 28 Calculated Reynolds numbers as a function of Sheet
only difference is the interpretation of the
basis weight for dewatering experiments at12 ms dwell time.
second order term in the former and the
deviation from unity of the exponent, n in the latter, which for flow through paper at moderate Reynolds
number should be interpreted as the effects of inertial forces due to deflections in the structure rather than
describing the effect of inertial forces due to turbulent eddies.
4.6
Flow Models for High Velocity Air Flow.
The experimental results obtained for flow of air through unbeaten and beaten paper sheets of different
grammage using vacuum dewatering equipment are shown in Figures 29 and 30 below. Figure 29 reports
data for a dwell time of 2 ms and Figure 30 for a dwell time of 12 ms. The data include a total of seven
different basis weights, three different pressure drops and four different degrees of beating (unbeaten and
beaten for 1000, 2000 and 3000 revolutions). One conclusion is that the flow rate is very much influenced
by the degree of beating so that increased beating of the pulp will lead to a decrease in the flow rate
through the paper, just as for the Bendtsen air permeance measurements reported earlier. Refining of pulp
improves the bonding ability of fibers, causing a variety of simultaneous changes in fibers such as;
internal fibrillation, external fibrillation, fiber shortening or cutting. Internal fibrillation of fibers improves
the flexibility and collapsibility of fibers, which further improves inter-fiber bonding. Another
explanation of low air flow by increasing degree of refining is that, refining makes the paper more dense
i.e. reduces the pore space between the fibers.
24 | P a g e
It is also clear from the data in Figures 29 and 30 that the parameter P/G seems to very relevant for
describing the flow rate so that the flow rate through the sample can be regarded as being a function of
P/G alone, as long as the degree of beating remains constant.
When it comes to the shape of the curves and the applicability of different types of mathematical models,
the results are not conclusive when comparing both graphs. The data in Figure 29 correspond to the
shorter dwell time are better described in terms of one of the models including inertial pressure drop and
compressibility effects, such as the one presented by Ramaswamy (Equation 10 and Figure 3). It can be
seen that the relationship between the pressure drop and the air flow rate is a non-linear expression and
that Darcy‟s law is not applicable. The possible explanation for such deviation is that Reynolds numbers
are high enough to correspond to laminar flow. The deviation from Darcy‟s law for paper from laminar to
turbulent conditions must be due to effect other than the transition. Polat explained the deviation from
linearity as being the result of inertial pressure losses as the air flow is subjected to rapid changes in the
direction of flow as it passes through the paper structure. For paper, which can be considered a very thin
porous bed, entrance and exit effects, i.e. irreversible losses due to acceleration and deceleration of the
fluid, are also likely to contribute significantly to the overall pressure drop.
Figure 29 show the experimental data of superficial velocity as a function of pressure drop obtained using
vacuum dewatering equipment; for unbeaten and beaten pulp samples at pressure drops of 20, 40 and 60
kPa for 2 ms dwell time.
80
Superficial Velocity, (m/s)
70
60
50
Unbeaten
40
Beaten 1000
30
Beaten 2000
20
Beaten 3000
10
0
0
1
2
3
4
(ΔP/G), (m2kPa/g)
Figure 29 Superficial air velocities as a function of pressure drop/ grammage for unbeaten and beaten pulp at 20, 40 and
60 kPa pressure drops for 2 ms dwell time experiments.
By assuming a constant density, behavior of such flows in which density does not vary significantly may
be simplified. This theory is termed as incompressible flow. However, the significant variations in density
can occur in many cases especially at higher velocities with large pressure changes. If a fluid exhibits
significant variations in density, its flow is considered to be compressible flow. Ramaswamy, (2003)
explained that during vacuum dewatering, the applied pressure differences can be quite high. Hence one
has to consider the compressibility effects of air as it passes through the sheet. It seems that the results
from the dewatering experiments are more likely similar to the model in which inertial forces are
dominating when taking compressibility into account.
25 | P a g e
The deviations from Darcy‟s law are smaller for the 12 ms dwell time data in Figure 30. The reason
behind this is not clear. However, the influence of any type of “offset” effects will be greater when
evaluating the flow rate according to Equations 16 and 17 based on only one dwell time. Due to the time
and effort involved it was not possible to evaluate every flow rate for a number of different dwell times as
was done for the data presented in Figure 22. However, one more way of evaluating the flow rate would
be using both dwell times so that flow rate is evaluated as
Equation 18
Figure 30 show the experimental data of superficial velocity as a function of pressure drop obtained using
vacuum dewatering equipment; for unbeaten and beaten pulp samples at pressure drops of 20, 40 and 60
kPa for 12 ms dwell time.
70
Superficial air velocity, (m/s)
60
50
40
Unbeaten
Beaten 1000
30
Beaten 2000
Beaten 3000
20
10
0
0
1
2
3
4
ΔP/G, (m2kPa/g)
Figure 30 Superficial air velocities as a function of pressure drop/ grammage for unbeaten and beaten pulp at 20, 40 and
60 kPa pressure drops for 12 ms dwell time experiments.
4.7
Leakage Experiments
The leakage was determined to investigate if air enters the vacuum tank without flowing through paper
sheets. The air flow results show that the higher leakages mean value i.e 0.44 kPa in terms of pressure
increase comes at 20 ms dwell time and 60 kPa pressure drop. The present study involves experiments at
2 ms and 12 ms dwell time. The higher mean value 0.36 kPa comes at 12 ms for 60 kPa pressure drop.
However the leakage values are so small that leakage can be neglected when calculating the air flow
26 | P a g e
Pressure Increase, kPa
through paper. Figure 31 show the leakage in terms of pressure increase as a function of dwell time for
20, 40 and 60 kPa pressure drops.
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
20 kPa
40 kPa
60 kPa
0
5
10
15
20
25
Vacuum dwell time, (ms)
Figure 31 Air leakage in terms of pressure increase as a function of dwell time at 20, 40 and 60 kPa.
5
CONCLUSION
Air flow through paper of basis weight (20-300) g/m2 was investigated experimentally. The Bendtsen air
permeance tester was used to investigate the air flow at pressure drop of 0.74, 1.47 and 2.20 kPa; while
the vacuum dewatering equipment was used to investigate air flow at pressure drop of 20, 40 and 60 kPa.
For Bendtsen air permeance tester, the majority of the experimental data from unbeaten and beaten pulp
sample agree rather well with Darcy‟s law i.e. there is a linear relationship between the volumetric flow
rate and the pressure drop/ grammage. However, for lower grammage sheet i.e. 20 g/m2 the obtained data
shows deviation from Darcy‟s law. The majority of the calculated Reynolds numbers are also in the
range, that they satisfy the applicability of Darcy‟s law. Further results from Bendtsen experiments are
that increased refining leads to less amount of air flow through the sheets. Also, it was found that a lower
grammage sheet accommodates large flow rates as compared to higher grammage sheets. However, when
comparing the vacuum dewatering experimental results with different theoretical models for flow through
porous media, it was found that the flow model in which inertial forces are dominating when taking
compressibility into account is applicable for experimental results at 2 ms dwell time. The deviation from
Darcy‟s law is smaller for the results of 12 ms dwell time experiments. The reason might be that the
influence of any type of “offset” effects will be greater when evaluating the flow rate according to
Equations 16 and 17 based on only one dwell time. Further results are that the parameter P/G, or
pressure drop divided by grammage seems to be very relevant for describing the flow rate so that the flow
rate through the sample can be regarded as being a function of P/G alone, as long as the degree of
beating remains constant. Also the air flow, by decreasing the sheet basis weight increases at first slightly,
then with a big jump between 50 g/m2 and 20 g/m2 sheets basis weight. It was also observed that the lower
basis weight sheet allows more air to penetrate in terms of pressure increase through it.
27 | P a g e
6
NOMENCLATURE
ΔP
Pressure difference
Pa
Fluid viscosity
Pa.s
k
Material Permeability
L
Thickness of Material
εg, εs and εw
volume fraction of gas solid and water
ρs
Fiber density
kgm-3/
ρw
Water density
kgm-3
G
Basis Weight
gm-2
a
Regression constant
n
exponent
L
Length of capillary
m
U
Superficial velocity
ms-1
α
Viscous Parameter
m-2
β
Inertial parameter
m-1
1
Inlet conditions
2
Outlet conditions
R
Gas constant
Re
Reynolds number
m
Mass flow rate,
Kg/sec
M
Molecular mass of air,
kg-mol-1
V
Volume of tank,
m3
T
Temperature,
K
Δt
Dwell time,
second
A
Cross section area,
m2
m
Pa m3/kg mol K
28 | P a g e
7
REFERENCES
Britt, K.W. and Unbehend, J.E, (1985), Water removal during paper formation, published in Tappi
journal, 68. (4), pp. 104-107.
Dullien, F.A.L, (1979), Porous media Fluid transport and pore structure.
Granevald, R., (2005), Doctoral thesis, Vacuum dewatering of low-grammage paper webs and fabrics.
Macdonald, I.F., El-Sayed, M.S., Mow, K, and Dullien, F.A.L., (1979), Flow through porous media-the
Ergun equation revisited., Published in Ind. Eng. Chem. Fundamental., Vol 18, No. 3.
Massey, B.S., (1990), Mechanics of fluids, Sixth edition.
Nordman, L. (1954), Laboratory investigation of water removal by a dynamic suction box, published in
Tappi journal, 37. (11), pp. 553-560.
Nilsson, L., (1996), Doctoral thesis, Some studies of the transport coefficients of pulp and paper.
Nilsson, L, and Stenstrom, S., (1996), A study of the permeability of pulp and paper, Published in
International journal for Multiphase Flow, Vol. 23, No 1, pp. 131-153.
Nilsson, L, Hussain, A. Abbas, A., (2011), Energy aspects on tissue production with the through air
drying technique, Paper accepted for publication in Nordic Drying Conference to be held in
Helsinki on June 19-21.
Polat, O., Crotogino, R.H., and Douglas, W.J.M (1992), Transport phenomena analysis of through drying
paper, published in Ind. Eng. Chem. Res, 21, 736-743.
Polat, O., Crotogino, R.H., Heiningen, V.A.R.P, and Douglas, W.J.M (1993), Permeability and specific
surface of paper, published in Journal of pulp and paper science: Vol. 19 No. 4.
Ramaswamy, S. (2003), Vacuum dewatering during paper manufacturing, Published in Drying
technology, Vol 21, No. 4, pp. 685-717.
Scheidegger, A.E. (1960), Physics of flow through porous media.
Tiller, F.M, Li, W.P, and Lee, J.B, (2001), Determination of the critical pressure drop for filtration of
super-compactible cakes, published in Water science and technology, Vol 44 No 10, pp. 171-176.
Weineisen, H, and Stenstrom, S., (2005), Modelling through drying of tissue-Effect of pore size
distribution on drying characteristics, Published in Drying technology, 23: 1909-1923.
Weineisen, H, Parent, L, Morrison, D, and Stenstrom, S. (2006), Experimental study of through-drying of
tissue at industrial conditions-Drying results and flow analysis, Published in 15th International
drying symposium, pp. 1026-1033.
Weineisen, H., Parent, L., Morrison, D., and Stenstrom, S., (2007), Through-drying of tissue at high
intensities- An experimental study, Published in journal of pulp and paper science, Vol. 33 No. 1.
Weineisen, H, (2007), Doctoral Thesis, Through drying of tissue paper.
29 | P a g e
8
ACKNOWLEDGEMENT
My supervisor, Dr. Lars Nilsson, is gratefully acknowledged for his skilled and confident guidance,
support and help throughout the studies and for his courage to take on this project. I am very grateful to
you for always have being available for discussion and support for my project. I really appreciate his
contribution during the last months when the structure of thesis was set.
All my colleagues at Karlstad University are greatly acknowledged for all their support and for creating a
pleasant atmosphere in which to work. I also express my gratitude to all staff at department of Chemical
Engineering at Karlstad University for always being helpful. Special thanks to Johan Kalender for
valuable discussion during data evaluation of dewatering equipment.
I would like to thanks Vesna for providing me the opportunity of using Bendtsen Air Permeance Tester at
StoraEnso Research Centre. I appreciated the pleasant cooperation we had during those measurements.
Staff at Karlstad Technical Centre is acknowledged for allowing me to perform Fiber testing analysis.
I would like to thanks Mikael Nilsson (Area Sales Manager) at Lorentzen & Wettre Skandinavien AB for
his technical comments regarding Bendtsen Air permeance tester.
I would also like to thanks my family for their prayers and their encouraging calls during my entire master
program. My friends for their support and encouragement are gratefully acknowledged.
30 | P a g e
9
APPENDIX
Table 3 Measured superficial velocities from Bendtsen Air Permeance Tester.
2.20 kPa
1.47kPa
0.74kPa
Pressure
Drop
Pulp
Sample
20
50
Unbeaten
2600
1000 rev
80
200
250
300
2100
100
120
150
180
Superficial air Velocity, (ml/min)
1700
1300
1180
800
600
480
420
400
2500
900
500
280
220
180
170
160
150
130
2000 rev
2200
500
320
220
190
170
110
90
80
60
3000 rev
2000
90
35
25
15
14
10
10
10
9
Unbeaten
8820
3580
3000
2500
2250
1550
1250
1000
900
800
1000 rev
8820
1800
1000
700
450
370
360
350
340
320
2000 rev
8820
700
480
350
290
210
150
110
90
70
3000 rev
4200
220
70
45
35
30
25
20
18
15
Unbeaten
1000 rev
2000 rev
3000 rev
8820
8820
8820
5850
5600
2600
1000
280
3550
1700
920
110
2880
1000
750
80
2650
800
500
55
2400
550
460
45
2000
480
350
35
1700
470
240
30
1500
460
140
30
1300
430
95
25
31 | P a g e
Table 4 Thickness values obtained for sheets made from hand sheet former.
Basis Weight (g/m2)
20
50
80
100
120
150
180
200
250
300
Unbeaten
Beaten 1000
Beaten 2000
Beaten 3000
Thickness, (μm)
47
112
164
207
247
309
361
396
498
530
41
86
145
189
218
244
301
326
401
462
37
84
124
154
187
219
255
300
341
401
34
74
105
138
148
197
230
254
300
309
Table 5 Thickness values obtained for sheets made from moified hand sheet former.
Basis weight (g/m2)
20
50
80
100
120
150
180
200
Unbeaten
67
194
279
283
331
396
434
484
Beaten1000 Beaten2000
Thickness, (μm)
61
58
173
164
233
226
281
252
325
277
375
330
428
387
479
417
Beaten3000
55
151
219
242
274
301
381
341
32 | P a g e
Table 6 Calculated Reynolds number for unbeaten pulp sheets, after measuring air flow using Bendtsen air
permeance tester.
Air Flow
(ml/min)
Air Flow
(m/s)
2500
900
500
280
220
180
170
160
150
130
8820
1800
1000
700
450
370
360
350
340
320
8820
2600
1700
1000
800
550
480
470
460
430
0.0417
0.0150
0.0083
0.0047
0.0037
0.0030
0.0028
0.0027
0.0025
0.0022
0.1470
0.0300
0.0167
0.0117
0.0075
0.0062
0.0060
0.0058
0.0057
0.0053
0.1470
0.0433
0.0283
0.0167
0.0133
0.0092
0.0080
0.0078
0.0077
0.0072
Hydrostatic
Air
diameter,
Density(δ)
Dh
kg/m3
(meter)
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
Air
Viscosity
(μ)
Pas
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
Reynolds
Number
0.050
0.018
0.010
0.006
0.004
0.004
0.003
0.003
0.003
0.003
0.178
0.036
0.020
0.014
0.009
0.007
0.007
0.007
0.007
0.006
0.178
0.052
0.034
0.020
0.016
0.011
0.010
0.009
0.009
0.009
33 | P a g e
Table 7 Calculated Reynolds number for beaten pulp sheets at 1000 revolution, after measuring air flow using
Bendtsen air permeance tester.
Air Flow
(ml/min)
Air Flow
(m/s)
2600
2100
1700
1300
1180
800
600
480
420
400
8820
3580
3000
2500
2250
1550
1250
1000
900
800
8820
5600
3550
2880
2650
2400
2000
1700
1500
1300
0.0433
0.0350
0.0283
0.0217
0.0197
0.0133
0.0100
0.0080
0.0070
0.0067
0.1470
0.0597
0.0500
0.0417
0.0375
0.0258
0.0208
0.0167
0.0150
0.0133
0.1470
0.0933
0.0592
0.0480
0.0442
0.0400
0.0333
0.0283
0.0250
0.0217
Hydrostatic
Air
Diameter,
Density(δ)
Dh
kg/m3
(meter)
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
Air
Viscosity
(μ)
Pas
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
Reynolds
Number
0.052
0.042
0.034
0.026
0.024
0.016
0.012
0.010
0.008
0.008
0.178
0.072
0.061
0.050
0.045
0.031
0.025
0.020
0.018
0.016
0.178
0.113
0.072
0.058
0.053
0.048
0.040
0.034
0.030
0.026
34 | P a g e
Table 8 Calculated Reynolds number for beaten pulp sheets at 2000 revolution, after measuring air flow using
Bendtsen air permeance tester.
Air Flow
(ml/min)
Air Flow
(m/s)
2200
500
320
220
190
170
110
90
80
60
8820
700
480
350
290
210
150
110
90
70
8820
1000
920
750
500
460
350
240
140
95
0.0367
0.0083
0.0053
0.0037
0.0032
0.0028
0.0018
0.0015
0.0013
0.0010
0.1470
0.0117
0.0080
0.0058
0.0048
0.0035
0.0025
0.0018
0.0015
0.0012
0.1470
0.0167
0.0153
0.0125
0.0083
0.0077
0.0058
0.0040
0.0023
0.0016
Hydrostatic
Air
diameter,
Density(δ)
Dh
kg/m3
(meter)
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
Air
Viscosity
(μ)
Pas
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
Reynolds
Number
0.044
0.010
0.006
0.004
0.004
0.003
0.002
0.002
0.002
0.001
0.178
0.014
0.010
0.007
0.006
0.004
0.003
0.002
0.002
0.001
0.178
0.020
0.019
0.015
0.010
0.009
0.007
0.005
0.003
0.002
35 | P a g e
Table 9 Calculated Reynolds number for beaten pulp sheets at 3000 revolution, after measuring air flow using
Bendtsen air permeance tester.
Air Flow
(ml/min)
Air Flow
(m/s)
2000
90
35
25
15
14
10
10
10
9
4200
220
70
45
35
30
25
20
18
15
5850
280
110
80
55
45
35
30
30
25
0.0333
0.0015
0.0006
0.0004
0.0003
0.0002
0.0002
0.0002
0.0002
0.0002
0.0700
0.0037
0.0012
0.0008
0.0006
0.0005
0.0004
0.0003
0.0003
0.0003
0.0975
0.0047
0.0018
0.0013
0.0009
0.0008
0.0006
0.0005
0.0005
0.0004
Hydrostatic
Air
diameter,
Density(δ)
Dh
kg/m3
(meter)
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
0.0000169
1.29
Air
Viscosity
(μ)
Pas
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
Reynolds
Number
0.040
0.002
0.001
0.001
0.000
0.000
0.000
0.000
0.000
0.000
0.085
0.004
0.001
0.001
0.001
0.001
0.001
0.000
0.000
0.000
0.118
0.006
0.002
0.002
0.001
0.001
0.001
0.001
0.001
0.001
36 | P a g e
Table 10 Mass flows for unbeaten and beaten pulp sheets to different revolutions, after measuring pressure
increase at 20 kPa using vacuum dewatering equipment at 2 ms dwell time.
(msec)
(sec)
(kPa)
(Pa)
2
0.002
1
1000
Molecular
mass of
air
M
(kg.mol-1)
0.029
0.3
Gas
Constant
R
(Nm.mol1
.K-1)
8.31
2
0.002
0.95
950
0.029
0.3
8.31
293
1.69724
2
0.002
0.8
800
0.029
0.3
8.31
293
1.42926
2
0.002
0.75
750
0.029
0.3
8.31
293
1.33993
2
0.002
0.65
650
0.029
0.3
8.31
293
1.16127
2
0.002
0.6
600
0.029
0.3
8.31
293
1.07194
2
0.002
0.5
500
0.029
0.3
8.31
293
0.89329
2
0.002
0.4
400
0.029
0.3
8.31
293
0.71463
2
0.002
0.9
900
0.029
0.3
8.31
293
1.60792
2
0.002
0.75
750
0.029
0.3
8.31
293
1.33993
2
0.002
0.7
700
0.029
0.3
8.31
293
1.25060
2
0.002
0.65
650
0.029
0.3
8.31
293
1.16127
2
0.002
0.6
600
0.029
0.3
8.31
293
1.07194
2
0.002
0.5
500
0.029
0.3
8.31
293
0.89329
2
0.002
0.4
400
0.029
0.3
8.31
293
0.71463
2
0.002
0.4
400
0.029
0.3
8.31
293
0.71463
2
0.002
0.8
800
0.029
0.3
8.31
293
1.42926
2
0.002
0.7
700
0.029
0.3
8.31
293
1.25060
2
0.002
0.65
650
0.029
0.3
8.31
293
1.16127
2
0.002
0.6
600
0.029
0.3
8.31
293
1.07194
2
0.002
0.55
550
0.029
0.3
8.31
293
0.98261
2
0.002
0.45
450
0.029
0.3
8.31
293
0.80396
2
0.002
0.35
350
0.029
0.3
8.31
293
0.62530
2
0.002
0.35
350
0.029
0.3
8.31
293
0.62530
2
2
2
2
2
2
2
2
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.3
600
550
500
450
400
350
300
300
0.029
0.029
0.029
0.029
0.029
0.029
0.029
0.029
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
8.31
8.31
8.31
8.31
8.31
8.31
8.31
8.31
293
293
293
293
293
293
293
293
1.07194
0.98261
0.89329
0.80396
0.71463
0.62530
0.53597
0.53597
Beaten 3000
Beaten 2000
Beaten 1000
Unbeaten Pulp
Dwell Time
Δt
Pressure
difference
ΔP
Tank
Volume
V
(m3)
Temperature
T
(K)
Mass Flow
m
(kg/sec)
293
1.78657
37 | P a g e
Table 11 Mass flows for unbeaten and beaten pulp sheets to different revolutions, after measuring pressure
increase at 40 kPa using vacuum dewatering equipment at 2 ms dwell time.
(msec)
(sec)
(kPa)
(Pa)
2
0.002
1.37
1370
Molecular
mass of
air
M
(kg.mol-1)
0.029
0.3
Gas
Constant
R
(Nm.mol1
.K-1)
8.31
2
0.002
1.24
1240
0.029
0.3
8.31
293
2.21535
2
0.002
1.04
1040
0.029
0.3
8.31
293
1.85804
2
0.002
0.98
980
0.029
0.3
8.31
293
1.75084
2
0.002
0.89
890
0.029
0.3
8.31
293
1.59005
2
0.002
0.65
650
0.029
0.3
8.31
293
1.16127
2
0.002
0.55
550
0.029
0.3
8.31
293
0.98261
2
0.002
0.5
500
0.029
0.3
8.31
293
0.89329
2
0.002
0.91
910
0.029
0.3
8.31
293
1.62578
2
0.002
0.85
850
0.029
0.3
8.31
293
1.51859
2
0.002
0.8
800
0.029
0.3
8.31
293
1.42926
2
0.002
0.75
750
0.029
0.3
8.31
293
1.33993
2
0.002
0.7
700
0.029
0.3
8.31
293
1.25060
2
0.002
0.65
650
0.029
0.3
8.31
293
1.16127
2
0.002
0.5
500
0.029
0.3
8.31
293
0.89329
2
0.002
0.4
400
0.029
0.3
8.31
293
0.71463
2
0.002
0.83
830
0.029
0.3
8.31
293
1.48286
2
0.002
0.8
800
0.029
0.3
8.31
293
1.42926
2
0.002
0.75
750
0.029
0.3
8.31
293
1.33993
2
0.002
0.7
700
0.029
0.3
8.31
293
1.25060
2
0.002
0.65
650
0.029
0.3
8.31
293
1.16127
2
0.002
0.6
600
0.029
0.3
8.31
293
1.07194
2
0.002
0.55
550
0.029
0.3
8.31
293
0.98261
2
0.002
0.45
450
0.029
0.3
8.31
293
0.80396
2
2
2
2
2
2
2
2
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.78
0.78
0.7
0.65
0.6
0.55
0.5
0.4
780
780
700
650
600
550
500
400
0.029
0.029
0.029
0.029
0.029
0.029
0.029
0.029
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
8.31
8.31
8.31
8.31
8.31
8.31
8.31
8.31
293
293
293
293
293
293
293
293
1.39353
1.39353
1.25060
1.16127
1.07194
0.98261
0.89329
0.71463
Beaten 3000
Beaten 2000
Beaten 1000
Unbeaten Pulp
Dwell Time
Δt
Pressure
difference
ΔP
Tank
Volume
V
(m3)
Temperature
T
(K)
Mass Flow
m
(kg/sec)
293
2.44760
38 | P a g e
Table 12 Mass flows for unbeaten and beaten pulp sheets to different revolutions, after measuring pressure
increase at 60 kPa using vacuum dewatering equipment at 2 ms dwell time.
(msec)
(sec)
(kPa)
(Pa)
2
0.002
1.39
1390
Molecular
mass of
air
M
(kg.mol-1)
0.029
0.3
Gas
Constant
R
(Nm.mol1
.K-1)
8.31
2
0.002
1.25
1250
0.029
0.3
8.31
293
2.23322
2
0.002
1.05
1050
0.029
0.3
8.31
293
1.87590
2
0.002
0.98
980
0.029
0.3
8.31
293
1.75084
2
0.002
0.75
750
0.029
0.3
8.31
293
1.33993
2
0.002
0.65
650
0.029
0.3
8.31
293
1.16127
2
0.002
0.55
550
0.029
0.3
8.31
293
0.98261
2
0.002
0.5
500
0.029
0.3
8.31
293
0.89329
2
0.002
0.94
940
0.029
0.3
8.31
293
1.67938
2
0.002
0.9
900
0.029
0.3
8.31
293
1.60792
2
0.002
0.85
850
0.029
0.3
8.31
293
1.51859
2
0.002
0.8
800
0.029
0.3
8.31
293
1.42926
2
0.002
0.75
750
0.029
0.3
8.31
293
1.33993
2
0.002
0.6
600
0.029
0.3
8.31
293
1.07194
2
0.002
0.5
500
0.029
0.3
8.31
293
0.89329
2
0.002
0.55
550
0.029
0.3
8.31
293
0.98261
2
0.002
0.85
850
0.029
0.3
8.31
293
1.51859
2
0.002
0.8
800
0.029
0.3
8.31
293
1.42926
2
0.002
0.75
750
0.029
0.3
8.31
293
1.33993
2
0.002
0.7
700
0.029
0.3
8.31
293
1.25060
2
0.002
0.65
650
0.029
0.3
8.31
293
1.16127
2
0.002
0.55
550
0.029
0.3
8.31
293
0.98261
2
0.002
0.5
500
0.029
0.3
8.31
293
0.89329
2
0.002
0.45
450
0.029
0.3
8.31
293
0.80396
2
2
2
2
2
2
2
2
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.5
800
750
700
650
600
550
500
500
0.029
0.029
0.029
0.029
0.029
0.029
0.029
0.029
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
8.31
8.31
8.31
8.31
8.31
8.31
8.31
8.31
293
293
293
293
293
293
293
293
1.42926
1.33993
1.25060
1.16127
1.07194
0.98261
0.89329
0.89329
Beaten 3000
Beaten 2000
Beaten 1000
Unbeaten Pulp
Dwell Time
Δt
Pressure
difference
ΔP
Tank
Volume
V
(m3)
Temperature
T
(K)
Mass Flow
m
(kg/sec)
293
2.48334
39 | P a g e
Table 13 Mass flows for unbeaten and beaten pulp sheets to different revolutions, after measuring pressure
increase at 20 kPa using vacuum dewatering equipment at 12 ms dwell time.
(msec)
(sec)
(kPa)
(Pas)
Molecular
mass of
air
M
(kg.mol-1)
12
0.012
2
2000
0.029
0.3
8.31
293
0.59552
12
0.012
1
1000
0.029
0.3
8.31
293
0.29776
12
0.012
0.8
800
0.029
0.3
8.31
293
0.23821
12
0.012
0.5
500
0.029
0.3
8.31
293
0.14888
12
0.012
0.7
700
0.029
0.3
8.31
293
0.20843
12
0.012
0.5
500
0.029
0.3
8.31
293
0.14888
12
0.012
0.4
400
0.029
0.3
8.31
293
0.11910
12
0.012
0.4
400
0.029
0.3
8.31
293
0.11910
12
0.012
1.9
1900
0.029
0.3
8.31
293
0.56575
12
0.012
0.9
900
0.029
0.3
8.31
293
0.26799
12
0.012
0.7
700
0.029
0.3
8.31
293
0.20843
12
0.012
0.6
600
0.029
0.3
8.31
293
0.17866
12
0.012
0.5
500
0.029
0.3
8.31
293
0.14888
12
0.012
0.5
500
0.029
0.3
8.31
293
0.14888
12
0.012
0.4
400
0.029
0.3
8.31
293
0.11910
12
0.012
0.3
300
0.029
0.3
8.31
293
0.08933
12
0.012
1.4
1400
0.029
0.3
8.31
293
0.41687
12
0.012
0.4
400
0.029
0.3
8.31
293
0.11910
12
0.012
0.8
800
0.029
0.3
8.31
293
0.23821
12
0.012
0.5
500
0.029
0.3
8.31
293
0.14888
12
0.012
0.5
500
0.029
0.3
8.31
293
0.14888
12
0.012
0.4
400
0.029
0.3
8.31
293
0.11910
12
0.012
0.3
300
0.029
0.3
8.31
293
0.08933
12
0.012
0.3
300
0.029
0.3
8.31
293
0.08933
12
0.012
1.3
1300
0.029
0.3
8.31
293
0.38709
12
0.012
0.7
700
0.029
0.3
8.31
293
0.20843
12
0.012
0.5
500
0.029
0.3
8.31
293
0.14888
12
0.012
0.5
500
0.029
0.3
8.31
293
0.14888
12
0.012
0.4
400
0.029
0.3
8.31
293
0.11910
12
0.012
0.3
300
0.029
0.3
8.31
293
0.08933
12
0.012
0.3
300
0.029
0.3
8.31
293
0.08933
12
0.012
0.2
200
0.029
0.3
8.31
293
0.05955
Beaten 3000
Beaten 2000
Beaten 1000
Unbeaten Pulp
Dwell Time
Δt
Pressure
difference
ΔP
Tank
Volume
V
(m3)
Gas
Constant
R
(Nm.mol1
.K-1)
Temperature
T
(K)
Mass Flow
m
(kg/sec)
40 | P a g e
Table 14 Mass flows for unbeaten and beaten pulp sheets to different revolutions, after measuring pressure
increase at 40 kPa using vacuum dewatering equipment at 12 ms dwell time.
(msec)
(sec)
(kPa)
(Pas)
Molecular
mass of
air
M
(kg.mol-1)
12
0.012
4.5
4500
0.029
0.3
8.31
293
1.33993
12
0.012
2.5
2500
0.029
0.3
8.31
293
0.74441
12
0.012
1.6
1600
0.029
0.3
8.31
293
0.47642
12
0.012
1.2
1200
0.029
0.3
8.31
293
0.35731
12
0.012
1.1
1100
0.029
0.3
8.31
293
0.32754
12
0.012
1
1000
0.029
0.3
8.31
293
0.29776
12
0.012
0.8
800
0.029
0.3
8.31
293
0.23821
12
0.012
0.8
800
0.029
0.3
8.31
293
0.23821
12
0.012
3.5
3500
0.029
0.3
8.31
293
1.04217
12
0.012
2
2000
0.029
0.3
8.31
293
0.59552
12
0.012
1.1
1100
0.029
0.3
8.31
293
0.32754
12
0.012
0.8
800
0.029
0.3
8.31
293
0.23821
12
0.012
0.8
800
0.029
0.3
8.31
293
0.23821
12
0.012
0.7
700
0.029
0.3
8.31
293
0.20843
12
0.012
0.6
600
0.029
0.3
8.31
293
0.17866
12
0.012
0.6
600
0.029
0.3
8.31
293
0.17866
12
0.012
2.5
2500
0.029
0.3
8.31
293
0.74441
12
0.012
1.2
1200
0.029
0.3
8.31
293
0.35731
12
0.012
0.9
900
0.029
0.3
8.31
293
0.26799
12
0.012
0.8
800
0.029
0.3
8.31
293
0.23821
12
0.012
0.7
700
0.029
0.3
8.31
293
0.20843
12
0.012
0.6
600
0.029
0.3
8.31
293
0.17866
12
0.012
0.6
600
0.029
0.3
8.31
293
0.17866
12
0.012
0.5
500
0.029
0.3
8.31
293
0.14888
12
0.012
2.2
2200
0.029
0.3
8.31
293
0.65508
12
0.012
1.1
1100
0.029
0.3
8.31
293
0.32754
12
0.012
0.8
800
0.029
0.3
8.31
293
0.23821
12
0.012
0.7
700
0.029
0.3
8.31
293
0.20843
12
0.012
0.5
500
0.029
0.3
8.31
293
0.14888
12
0.012
0.6
600
0.029
0.3
8.31
293
0.17866
12
0.012
0.5
500
0.029
0.3
8.31
293
0.14888
12
0.012
0.6
600
0.029
0.3
8.31
293
0.17866
Beaten 3000
Beaten 2000
Beaten 1000
Unbeaten Pulp
Dwell Time
Δt
Pressure
difference
ΔP
Tank
Volume
V
(m3)
Gas
Constant
R
(Nm.mol1
.K-1)
Temperature
T
(K)
Mass Flow
m
(kg/sec)
41 | P a g e
Table 15 Mass flows for unbeaten and beaten pulp sheets to different revolutions, after measuring pressure
increase at 60 kPa using vacuum dewatering equipment at 12 ms dwell time.
(msec)
(sec)
(kPa)
(Pas)
Molecular
mass of
air
M
(kg.mol-1)
12
0.012
8
8000
0.029
0.3
8.31
293
2.38210
12
0.012
5.4
5400
0.029
0.3
8.31
293
1.60792
12
0.012
3.8
3800
0.029
0.3
8.31
293
1.13150
12
0.012
2.4
2400
0.029
0.3
8.31
293
0.71463
12
0.012
2
2000
0.029
0.3
8.31
293
0.59552
12
0.012
1.6
1600
0.029
0.3
8.31
293
0.47642
12
0.012
1
1000
0.029
0.3
8.31
293
0.29776
12
0.012
0.8
800
0.029
0.3
8.31
293
0.23821
12
0.012
4
4000
0.029
0.3
8.31
293
1.19105
12
0.012
2
2000
0.029
0.3
8.31
293
0.59552
12
0.012
1.5
1500
0.029
0.3
8.31
293
0.44664
12
0.012
1.2
1200
0.029
0.3
8.31
293
0.35731
12
0.012
1
1000
0.029
0.3
8.31
293
0.29776
12
0.012
0.8
800
0.029
0.3
8.31
293
0.23821
12
0.012
0.7
700
0.029
0.3
8.31
293
0.20843
12
0.012
0.6
600
0.029
0.3
8.31
293
0.17866
12
0.012
3.6
3600
0.029
0.3
8.31
293
1.07194
12
0.012
1.6
1600
0.029
0.3
8.31
293
0.47642
12
0.012
1.2
1200
0.029
0.3
8.31
293
0.35731
12
0.012
1.2
1200
0.029
0.3
8.31
293
0.35731
12
0.012
1
1000
0.029
0.3
8.31
293
0.29776
12
0.012
0.8
800
0.029
0.3
8.31
293
0.23821
12
0.012
0.7
700
0.029
0.3
8.31
293
0.20843
12
0.012
0.5
500
0.029
0.3
8.31
293
0.14888
12
0.012
3
3000
0.029
0.3
8.31
293
0.89329
12
0.012
1.4
1400
0.029
0.3
8.31
293
0.41687
12
0.012
1.2
1200
0.029
0.3
8.31
293
0.35731
12
0.012
1
1000
0.029
0.3
8.31
293
0.29776
12
0.012
0.8
800
0.029
0.3
8.31
293
0.23821
12
0.012
0.8
800
0.029
0.3
8.31
293
0.23821
12
0.012
0.7
700
0.029
0.3
8.31
293
0.20843
12
0.012
0.5
500
0.029
0.3
8.31
293
0.14888
Beaten 3000
Beaten 2000
Beaten 1000
Unbeaten Pulp
Dwell Time
Δt
Pressure
difference
ΔP
Tank
Volume
V
(m3)
Gas
Constant
R
(Nm.mol1
.K-1)
Temperature
T
(K)
Mass Flow
m
(kg/sec)
42 | P a g e
Beaten 3000
Beaten 2000
Beaten 1000
Unbeaten Pulp
Table 16 Superficial air velocity for unbeaten and beaten pulp sheets to different revolutions, after measuring
mass flow at 20 kPa using vacuum dewatering equipment at 2 ms dwell time.
Mass Air Flow
(kg/sec)
1.79
1.70
1.43
1.34
1.16
1.07
0.89
0.71
1.61
1.34
1.25
1.16
1.07
0.89
0.71
0.71
1.43
1.25
1.16
1.07
0.98
0.80
0.63
0.63
1.07
0.98
0.89
0.80
0.71
0.63
0.54
0.54
Air Density
(kg/m3)
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
Cross
Section
Area
(m2)
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
Air Velocity
(m/sec)
49.462
46.989
39.570
37.097
32.150
29.677
24.731
19.785
44.516
37.097
34.623
32.150
29.677
24.731
19.785
19.785
39.570
34.623
32.150
29.677
27.204
22.258
17.312
17.312
29.677
27.204
24.731
22.258
19.785
17.312
14.839
14.839
43 | P a g e
Beaten 3000
Beaten 2000
Beaten 1000
Unbeaten Pulp
Table 17 Superficial air velocity for unbeaten and beaten pulp sheets to different revolutions, after measuring
mass flow at 40 kPa using vacuum dewatering equipment at 2 ms dwell time.
Mass Air Flow
(kg/sec)
2.45
2.22
1.86
1.75
1.59
1.16
0.98
0.89
1.63
1.52
1.43
1.34
1.25
1.16
0.89
0.71
1.48
1.43
1.34
1.25
1.16
1.07
0.98
0.80
1.39
1.39
1.25
1.16
1.07
0.98
0.89
0.71
Air Density
(kg/m3)
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
Cross Section
Area
(m2)
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
Air Velocity
(m/sec)
67.763
61.333
51.441
48.473
44.021
32.150
27.204
24.731
45.011
42.043
39.570
37.097
34.623
32.150
24.731
19.785
41.054
39.570
37.097
34.623
32.150
29.677
27.204
22.258
38.580
38.580
34.623
32.150
29.677
27.204
24.731
19.785
44 | P a g e
Beaten 3000
Beaten 2000
Beaten 1000
Unbeaten Pulp
Table 18 Superficial air velocity for unbeaten and beaten pulp sheets to different revolutions, after measuring
mass flow at 60 kPa using vacuum dewatering equipment at 2 ms dwell time.
Mass Air Flow
(kg/sec)
2.48
2.23
1.88
1.75
1.34
1.16
0.98
0.89
1.68
1.61
1.52
1.43
1.34
1.07
0.89
0.98
1.52
1.43
1.34
1.25
1.16
0.98
0.89
0.80
1.43
1.34
1.25
1.16
1.07
0.98
0.89
0.89
Air Density
(kg/m3)
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
Cross
Section
Area
(m2)
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
Air Velocity
(m/sec)
68.752
61.828
51.935
48.473
37.097
32.150
27.204
24.731
46.494
44.516
42.043
39.570
37.097
29.677
24.731
27.204
42.043
39.570
37.097
34.623
32.150
27.204
24.731
22.258
39.570
37.097
34.623
32.150
29.677
27.204
24.731
24.731
45 | P a g e
Beaten 3000
Beaten 2000
Beaten 1000
Unbeaten Pulp
Table 19 Superficial air velocity for unbeaten and beaten pulp sheets to different revolutions, after measuring
mass flow at 20 kPa using vacuum dewatering equipment at 12 ms dwell time.
Mass Air Flow
(kg/sec)
Air Density
(kg/m3)
0.60
0.30
0.24
0.15
0.21
0.15
0.12
0.12
0.57
0.27
0.21
0.18
0.15
0.15
0.12
0.09
0.42
0.12
0.24
0.15
0.15
0.12
0.09
0.09
0.39
0.21
0.15
0.15
0.12
0.09
0.09
0.06
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
Cross
Section
Area
(m2)
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
Air Velocity
(m/sec)
16.487
8.244
6.595
4.122
5.771
4.122
3.297
3.297
15.663
7.419
5.771
4.946
4.122
4.122
3.297
2.473
11.541
3.297
6.595
4.122
4.122
3.297
2.473
2.473
10.717
5.771
4.122
4.122
3.297
2.473
2.473
1.649
46 | P a g e
Beaten 3000
Beaten 2000
Beaten 1000
Unbeaten Pulp
Table 20 Superficial air velocity for unbeaten and beaten pulp sheets to different revolutions, after measuring
mass flow at 40 kPa using vacuum dewatering equipment at 12 ms dwell time.
Mass Air Flow
(kg/sec)
Air Density
(kg/m3)
Cross Section Area
(m2)
Air Velocity
(m/sec)
1.34
0.74
0.48
0.36
0.33
0.30
0.24
0.24
1.04
0.60
0.33
0.24
0.24
0.21
0.18
0.18
0.74
0.36
0.27
0.24
0.21
0.18
0.18
0.15
0.66
0.33
0.24
0.21
0.15
0.18
0.15
0.18
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
37.097
20.609
13.190
9.892
9.068
8.244
6.595
6.595
28.853
16.487
9.068
6.595
6.595
5.771
4.946
4.946
20.609
9.892
7.419
6.595
5.771
4.946
4.946
4.122
18.136
9.068
6.595
5.771
4.122
4.946
4.122
4.946
47 | P a g e
Beaten 3000
Beaten 2000
Beaten 1000
Unbeaten Pulp
Table 21 Superficial air velocity for unbeaten and beaten pulp sheets to different revolutions, after measuring
mass flow at 60 kPa using vacuum dewatering equipment at 12 ms dwell time.
Mass Air Flow
(kg/sec)
Air Density
(kg/m3)
Cross
Section Area
(m2)
Air Velocity
(m/sec)
2.38
1.61
1.13
0.71
0.60
0.48
0.30
0.24
1.19
0.60
0.45
0.36
0.30
0.24
0.21
0.18
1.07
0.48
0.36
0.36
0.30
0.24
0.21
0.15
0.89
0.42
0.36
0.30
0.24
0.24
0.21
0.15
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
0.028
65.950
44.516
31.326
19.785
16.487
13.190
8.244
6.595
32.975
16.487
12.366
9.892
8.244
6.595
5.771
4.946
29.677
13.190
9.892
9.892
8.244
6.595
5.771
4.122
24.731
11.541
9.892
8.244
6.595
6.595
5.771
4.122
48 | P a g e
Table 22 Calculated Reynolds numbers for unbeaten and beaten pulp sheets to different revolution, after
measuring air flow at 20 kPa using vacuum dewatering equipment at 2 ms dwell time.
Air Flow
(m/s)
49.4621
46.9890
39.5697
37.0966
32.1504
29.6773
24.7311
19.7849
44.5159
37.0966
34.6235
32.1504
29.6773
24.7311
19.7849
19.7849
39.5697
34.6235
32.1504
29.6773
27.2042
22.2580
17.3117
17.311748
29.677282
27.204175
24.731068
22.257961
19.784855
17.311748
14.838641
14.838641
Hydrostatic
Air
Diameter,
Density(δ)
Dh
kg/m3
(meter)
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
Air
Viscosity
(μ)
Pas
Reynolds
Number
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
59.907
56.912
47.926
44.930
38.939
35.944
29.953
23.963
53.916
44.930
41.935
38.939
35.944
29.953
23.963
23.963
47.926
41.935
38.939
35.944
32.949
26.958
20.967
20.967
35.944
32.949
29.953
26.958
23.963
20.967
17.972
17.972
49 | P a g e
Table 23 Calculated Reynolds numbers for unbeaten and beaten pulp sheets to different revolution, after
measuring air flow at 40 kPa using vacuum dewatering equipment at 2 ms dwell time.
Air Flow
(m/s)
67.7631
61.3330
51.4406
48.4729
44.0213
32.1504
27.2042
24.7311
45.0105
42.0428
39.5697
37.0966
34.6235
32.1504
24.7311
19.7849
41.0536
39.5697
37.0966
34.6235
32.1504
29.6773
27.2042
22.257961
38.580466
38.580466
34.623496
32.150389
29.677282
27.204175
24.731068
19.784855
Hydrostatic
Air
Air
Diameter,
Viscosity Reynolds
Density(δ)
Dh
(μ)
Number
kg/m3
(meter)
Pas
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
82.072
74.285
62.303
58.709
53.317
38.939
32.949
29.953
54.515
50.921
47.926
44.930
41.935
38.939
29.953
23.963
49.723
47.926
44.930
41.935
38.939
35.944
32.949
26.958
46.727
46.727
41.935
38.939
35.944
32.949
29.953
23.963
50 | P a g e
Table 24 Calculated Reynolds numbers for unbeaten and beaten pulp sheets to different revolution, after
measuring air flow at 60 kPa using vacuum dewatering equipment at 2 ms dwell time.
Air Flow
(m/s)
68.7524
61.8277
51.9352
48.4729
37.0966
32.1504
27.2042
24.7311
46.4944
44.5159
42.0428
39.5697
37.0966
29.6773
24.7311
27.2042
42.0428
39.5697
37.0966
34.6235
32.1504
27.2042
24.7311
22.257961
39.569709
37.096602
34.623496
32.150389
29.677282
27.204175
24.731068
24.731068
Hydrostatic
Air
Diameter,
Density(δ)
Dh
kg/m3
(meter)
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
Air
Viscosity
(μ)
Pas
Reynolds
Number
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
83.271
74.884
62.902
58.709
44.930
38.939
32.949
29.953
56.312
53.916
50.921
47.926
44.930
35.944
29.953
32.949
50.921
47.926
44.930
41.935
38.939
32.949
29.953
26.958
47.926
44.930
41.935
38.939
35.944
32.949
29.953
29.953
51 | P a g e
Table 25 Calculated Reynolds numbers for unbeaten and beaten pulp sheets to different revolution, after
measuring air flow at 20 kPa using vacuum dewatering equipment at 12 ms dwell time.
Air Flow
(m/s)
16.4874
8.2437
6.5950
4.1218
5.7706
4.1218
3.2975
3.2975
15.6630
7.4193
5.7706
4.9462
4.1218
4.1218
3.2975
2.4731
11.5412
3.2975
6.5950
4.1218
4.1218
3.2975
2.4731
10.716796
5.7705826
4.1218447
4.1218447
3.2974758
2.4731068
2.4731068
1.6487379
Hydrostatic
Air
Diameter,
Density(δ)
Dh
kg/m3
(meter)
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
Air
Viscosity
(μ)
Pas
Reynolds
Number
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
19.969
9.984
7.988
4.992
6.989
4.992
3.994
3.994
18.971
8.986
6.989
5.991
4.992
4.992
3.994
2.995
13.978
3.994
7.988
4.992
4.992
3.994
2.995
12.980
6.989
4.992
4.992
3.994
2.995
2.995
1.997
52 | P a g e
Table 26 Calculated Reynolds numbers for unbeaten and beaten pulp sheets to different revolution, after
measuring air flow at 40 kPa using vacuum dewatering equipment at 12 ms dwell time.
Air Flow
(m/s)
37.0966
20.6092
13.1899
9.8924
9.0681
8.2437
6.5950
6.5950
28.8529
16.4874
9.0681
6.5950
6.5950
5.7706
4.9462
4.9462
20.6092
9.8924
7.4193
6.5950
5.7706
4.9462
4.9462
18.136117
9.0680583
6.5949515
5.7705826
4.1218447
4.9462136
4.1218447
4.9462136
Hydrostatic
Air
Air
Diameter,
Viscosity Reynolds
Density(δ)
Dh
(μ)
Number
kg/m3
(meter)
Pas
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
44.930
24.961
15.975
11.981
10.983
9.984
7.988
7.988
34.946
19.969
10.983
7.988
7.988
6.989
5.991
5.991
24.961
11.981
8.986
7.988
6.989
5.991
5.991
21.966
10.983
7.988
6.989
4.992
5.991
4.992
5.991
53 | P a g e
Table 27 Calculated Reynolds numbers for unbeaten and beaten pulp sheets to different revolution, after
measuring air flow at 60 kPa using vacuum dewatering equipment at 12 ms dwell time.
Air Flow
(m/s)
65.9495
44.5159
31.3260
19.7849
16.4874
13.1899
8.2437
6.5950
32.9748
16.4874
12.3655
9.8924
8.2437
6.5950
5.7706
4.9462
29.6773
13.1899
9.8924
9.8924
8.2437
6.5950
5.7706
24.731068
11.541165
9.8924273
8.2436894
6.5949515
6.5949515
5.7705826
4.1218447
Hydrostatic
Air
Diameter,
Density(δ)
Dh
kg/m3
(meter)
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
0.0000169
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
1.29
Air
Viscosity
(μ)
Pas
Reynolds
Number
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
0.000018
79.876
53.916
37.941
23.963
19.969
15.975
9.984
7.988
39.938
19.969
14.977
11.981
9.984
7.988
6.989
5.991
35.944
15.975
11.981
11.981
9.984
7.988
6.989
29.953
13.978
11.981
9.984
7.988
7.988
6.989
4.992
54 | P a g e