Displacement of Fibre Network

MASTER'S THESIS

Displacement of Fibre Network

Andreas Emilsson 2016

Master of Science in Engineering Technology Engineering Physics and Electrical Engineering

Luleå University of Technology

Department of Engineering Sciences and Mathematics

Master thesis - Displacement of fibre network

Andreas Emilsson June 14, 2016

Lule˚ a University of Technology

Engineering physics and electrical Engineering

Abstract

The dewatering process is very important in pulp making industry since the process

recycle chemicals and therefore reduce the production cost. The process also improves

the quality of the final product as well as there is some environmental aspects. This

thesis is done for Valmet. The objective is to simulate compression of fibres due to

hydraulic pressure. Simulations are made using previous data obtained for different

pulps. The simulations are executed using Ansys CFX 16.2. Multiphase flow was

used where the phases are water and fibre. Modifications of the standard CFX solver

has been made in order to stabilize the solver and obtaining results. Results of two

different pulps are presented and the result seems accurate and the forces seems to

be in balance. Terazaghi’s principle is full filled as well. One main conclusion is that

the permeability and solid pressure has a huge effect on the final results and it is

very important that the material constants is accurate. The adjustment of solver

parameters is also important for retrieving force balance.

Acknowledgements

First of all I would like to thank my supervisor in Sundsvall Tomas Vikstr¨ om for his great support and guidance through the thesis work. I would also like to thank my collegians in Sundsvall for their kindness and inviting attitude. I have always felt appreciated which have led too that I will think back to Sundsvall as a great place with a lot of great memories.

A special thanks to my new and hopefully long lasting friendship with Adam J¨ onsson for introducing me to his friends and spending time to know me out side of work. I haven’t had this much fun as I had with you during this time in Sundsvall for a long time.

I would also like to thank my Anton Burman, Jan-Erik Kitok and Gustaf Alnersson for their kindness of lending me a spot in their office during my days in Lule˚ a. Also I would like to thank Gunnar Hellstr¨ om for his support and supervision in Lule˚ a.

Lastly I would like to thank friends and family for their support during less motivating

times and also for their belief in this thesis work.

Contents

1 Introduction 1

1.1 Chemical process of pulp . . . . 1

1.2 Fibres . . . . 3

1.2.1 Cellulose . . . . 3

1.2.2 Hemicellulose . . . . 4

1.2.3 Lignin . . . . 4

1.3 Pulp . . . . 4

1.4 Objective . . . . 4

1.5 Earlier work . . . . 5

2 Theory 8 2.1 Fluid mechanics . . . . 8

2.2 Porous media . . . . 9

2.3 Compressibility . . . . 10

2.4 Multiphase . . . . 11

2.4.1 Eulerian-Eulerian model . . . . 12

2.4.2 Lagrangian particle tracking model . . . . 13

2.5 Drag models . . . . 14

2.6 Interphase turbulence dispersion models . . . . 16

2.6.1 Favre average drag model . . . . 16

2.6.2 Lopez de Bertodano model . . . . 17

2.7 Solid pressure models . . . . 17

3 Method 19 3.1 Fibre Material . . . . 19

3.1.1 Drag force . . . . 19

3.2 Geometry and mesh . . . . 19

3.3 Boundary conditions and initialization . . . . 20

4 Results 24

5 Discussion and Conclusion 28

6 Future work 29

7 Appendix 30

1 Introduction

This thesis work is done for Valmet and the aim with the thesis is to study the flow during the washing process. The material models for solid pressure and drag coef- ficients was retrieved from literature. The purpose of the work is to get a better understanding of the compression process which can be used for e.g. optimization of the machine and execute the washing process faster which will result in reducing the production costs.

Limitations has been done by solving it in 2-d instead of 3-d. The limitation is made because of the complexity of the problem as well as saving computational resources. Since only the numerical method of the solver is of importance is it un- necessary with a complex geometry. Another assumption which is made is that the fibres in the process are of the same size, this will lower the complexity and result in faster simulations. The permeability model also assume that the fibres are equally distributed across the domain.

The methodology of this work is to begin with a literature study to obtain better understanding of the problem before implementing the problem into Ansys CFX 16.2.

1.1 Chemical process of pulp

The overall description of a typical pulp mill can be seen in Fig. 1.

Figure 1: Picture of the pulp mill.

Where the process is described step by step below.

• First step of the pulp process is to debark the trees and cut the log down into small chips. Where the size of the chips is approximately 20x30x10mm.

• Next step is to digest the chips with a mixture of heat, water and liquor to re- solve the lignin and to break down the chips into fibres. The size of the fibres are approximately 1mm long and has a diameter around 20µm. Chemicals often used are sodium sulphite or sodium hydroxide. The mixture will not only affect the lignin but the cellulose and hemicellulose as well. After the cooking process most of the lignin is dissolved and some of the hemicellulose and a little of the cellulose.

• The mixture is then pumped into the washing process. During the washing process the dissolved substances is removed from the cooking mixture. The mixture consist mostly of lignin, hemicellulose and chemicals. The chemicals are recycled and reused to cut the production costs and to save on the environment. Dissolved lignin and hemicellulose can be burnt down to heat up steam that is used in the cooking pro- cess. The remaining mixture after the washing process is now a mixture of clean water with cellulose and hemicellulose.

• A small quantity of lignin can still be left and can then decolour the pulp, hence

the mixture is bleached in the last and final step. The mixture must be bleached if it

is required that the final product is colourless. The bleaching in done by removing as

much as possible of the remaining quantities of lignin without harming the cellulose and hemicellulose. The bleaching process is done by mixing the pulp with sodium hydroxide and wash it with alkali water. This process is repeated until the required brightness is obtained.

1.2 Fibres

Figure 2: Picture of fibres.

Wood is built up by fibres which consist of cellulose, hemicellulose and lignin. The composition differs for each wood type. Fibres looks like straws with a fibre wall thickness which is dependent at what time of the summer it is harvested. The fibre wall thickness is thicker for hardwood than for softwood and the fibre wall of the fibres will be thicker in the summer and thinner in the spring. The length of the fibres will also vary depending on which type of species it is. Softwood will have a length of approximately 3mm and hardwood will have a length around 1 mm. The length of the fibre will differ as well as the thickness of the wall, it will be longer in the summer and shorter in the spring. The total diameter of the hardwood fibre is 30µm and for softwood 20µm. The diameter of the fibre will be larger in the spring and smaller in the summer.[1]

1.2.1 Cellulose

Cellulose is a molecule chain built up by a large number of glucose molecules. The

cellulose represents approximately 40% of the chips, and it is the substance which

creates the structure and the stability of the chips. Cellulose is not resolvable in

water and it is colour less. The resulting pulp is manly cellulose and hemicellulose.

1.2.2 Hemicellulose

Hemicellulose is built up differently from cellulose, it is built up by different carbo- hydrates molecules instead of glucose molecules. The hemicellulose works like the glue between the cellulose molecules in the fibre wall. An other difference between hemicellulose and cellulose is that hemicellulose is resolvable in water, which cellulose is not.

1.2.3 Lignin

Lignin molecule is built up by a three dimensional network and is not resolvable in water unless it reacts in high temperature with sour or alkaline chemicals, such as sulphur. Lignin works like glue between the fibres and keeps them together. Lignin loosens up in high temperatures, around 150

C − 170

C, which is useful when the wood is broken down into fibres in the mechanical processes[2]. Lignin is the substance which is dissolved in the pulp making industry due to the de strengthening and de colouring of the final product. But the lignin can not be totally dissolved because of the complexity to make the substance resolvable. Lignin is not colourless, it has a yellow colour and therefore is bleaching one important step in the pulp manufacturing process.

1.3 Pulp

Pulp is a network of entangled fibres which in theory is evenly distributed to obtain as good quality as possible on the final product. The network will act like a spring and a damper when compressed due to fibre fibre contact as well as a damper from energy losses due to friction between the phases. The consistency of the fibre suspension at 4% fibre is close to the consistency of oatmeal with too much water. A concentration of 50% will act like wood. After the washing machine the dryness of the pulp is 30 − 40% so the resulting product after the washing process is very dry.

1.4 Objective

The tasks is to see if it is possible to simulate one part of the washing process of pulp

in a washing press using commercial software. The software used is ANSYS CFX and

a figure of the washing machine can be seen in Fig. 3.

Figure 3: Figure over a washing press.

1.5 Earlier work

Earlier work in this field has been done, not on the exact same problem, but on a problem which has some similarities. Here a 3-d simulation of compression of a fibre network has been made in CFD fluent during 2007[3]. Were the objectives was to create a geometry and compress the fibre suspension in order to imitate a hot calen- dering process. The simulations showed good agreement with empirical permeability models as well as earlier models derived in other works. Assumptions during the work was made to simplify the complexity of the task. Where one simplification was that the fibres looked like blocks with a cross section geometry like a square. When the fibres were compressed they bent like simple models.

Studies has been made of the flow through and deformation of the fiber network [4]. Where objectives was to describe and improve already existing pressure models when washing pulp. To be able to describe the pressure the author was forced to derive some material properties such as permeability and solid pressure. Studies were made of the permeability in two cases. The first case determined the permeability with air. The permeability is derived using Darcy’s law, which can be seen in Eq. 1.

δP

δx = −µ · U

k . (1)

µ is the viscosity of the fluid, k is the permeability, U is the average velocity of the field and P is the pressure. Darcy’s law is rewritten so it was expressed as mass flow using

P · v · A

= ˙ m · R · T. (2)

v is the superficial velocity, A

is the area of the cavity, ˙ m is the mass flow, R is

the gas constant and T is the temperature. He obtained an empirical model for the

permeability of air, Eq. 3

k = 2µRT h

A

(P

− P

) . (3)

h is the height of the experiment domain. It was also found that the fibre sus- pension is anisotropic which means that the fibre suspension has different physical properties in different directions.

Empirical constants , C

, C

and S

are derived for the permeability equation,4, which is derived by Ingmansson, Andrews et al.[5].

k = (C

· S

· (1 − )

· (1 + C

· (1 − )

))

(4) where C

, C

and S

are varying for different kinds of pulp and  is the porosity of the matrix. Figure over different kinds of permeability graphs can be seen in Fig. 4.

Figure 4: Figure over example of different permeability graphs for different pulps.

Where the y-axis is logarithmic and the volume fraction is displayed on the x-axis.

It is clear from the figure that the data from literature shows of high volatility. The author also derived an empirical model for the solid pressure which occur when a fibre suspension is compressed. The equation derived can be seen in Eq. 5.

P

= E

φ

(1 − φ)

(5)

where E

,E

and E

are empirical constants and graphs over different kinds of solid pressure graphs for different pulps can be seen in Fig.5.

Figure 5: Figure over different solid pressure graphs for different pulps.

Y-axis shows the solid pressure on a logarithmic scale and the volume fraction is showed on the x-axis.

Another thesis work has been done where the objective with the work was to analyse the behaviour of the fibres when they are compressed due to washing. Open- FOAM was used to analyse the problem with a Eulerian-Eulerian multiphase model.

The models used for describing compressibility and permeability of the fibres was

given from Valmet. Simulations were done for 1-d, 2-d and 3-d geometries. Conver-

gence test showed that the mesh cells size of 100 to 500 cells are good enough. A

time step test was also done with 200 cells where the conclusion was that a step size

of t = 1 · 10

s results in a decent result.

2 Theory

2.1 Fluid mechanics

In fluid dynamicsyou study the behaviour of gas or liquid when applying a pressure or force to the system. The flows are described with Navier-Stokes equation which can be seen in Eq. 6

ρ ∂u

∂t + ρ(u · ∇)u = −∇p + F + µ∇

u (6)

where u is the velocity vector and F is the outer forces acting on the fluid i.e.

gravity, pressure differences or shear stresses. Where the terms are:

•ρ

is describing the transient behaviour in the system.

•ρ(u · ∇)u is the term which is describing the convection in the system.

•∇p is describing the pressure difference.

•µ∇

u is describing the viscous forces.

•F is describing the outer forces such as pressure, gravity and drag forces. In our case is it describing the solid pressure and drag forces due to interaction between phases.

An other important equation is the continuity equation which says that the mass in the system is conserved. The continuity equation can be seen in Eq. 7

dρ

dt + ∇ · (ρu) = 0 (7)

where

•

is describing the compressibility of the phase.

•∇ · (ρu) is describing the convective flux of the phase.

The behaviour of the fluid or gas can be described in two states. Turbulent or laminar state. In which state they are in is dependent on the Reynolds number. The Reynolds number is calculated using Eq. 8

Re = ρU L

µ (8)

The transition region is also depending on the smoothness of the surface closest to

the flow and the transition region between turbulence to laminar can therefore vary

a lot. The transition region is often between Re = 1000 to Re = 10000 [6]. One

method to solve fluid dynamics problem is to use a computational fluid dynamics program e.g. ANSYS. Programs as ANSYS discretizate the Navier-Stokes equation, Eq. 6, and the continuity equation, Eq. 7, into a number of points depending on the mesh. It solves the equation in each node and average it into the center of the mesh cell. Therefore is the quality of the mesh an important aspect when solving problems.

A poor mesh will result in poor results or even wrong results.

2.2 Porous media

The definition of a porous media can briefly be described as a solid with internal canals, the canals are continuous irregularly distributed pathways inside the solid.

The solid is often refereed to as a solid matrix and the canals pore space or void space [7]. Some examples of where porous media can be encountered are soil science, soil mechanics, chemical engineering, filtration and in our case, fibre networks.

A parameter of the magnitude of the porosity in a domain is introduced. The param- eter is  and it is defined as

 = 1 − V

. (9)

where V

is the volume fraction of the solid in the domain. Porosity will affect the permeability of the domain i.e. how easily the flow will flow through the medium.

High permeability will result in less friction so the liquid will flow through the medium easier. The permeability, k, can be derived from Darcy’s law which can be seen in Eq. 1 and one of the most broadly known equation for the permeability is the Kozeny- Carman equation and it is defined as Eq. 10,

K = (a · S

)

· 

(1 − )

(10)

where the constant a is the Kozeny constant and S

is a constant for the specific surface for each type of fibre. But it is found that the Kozeny-Carman equation describes the permeability poor when the porosity becomes too high [8]. One study shows that Kozeny-Carman is valid for a porosity ranging from 0.2 <  < 0.7 [9]. The equation of permeability was therefore modified into Eq. 4 and this is the permeabil- ity equation used for this case.

Tortuosity can also be an important factor when describing a porous media. Tor-

tuosity is the ratio between the average pore length and the length of the porous

medium. So it is a factor of how the spores twist inside the porous medium. If the

tortuoisity is equal to one, then the spores are like straight tubes inside the porous

media and as tortuoisity increases the pores become more and more twisted inside the

medium. The tortuosity will effect the terms in Navier-Stoke equation by influencing

the velocity gradient term, ∇u. As a result of this will the inertia of the fluid will be

more significant.[10]

2.3 Compressibility

When compressing a fibre network the interplay between the fluid and the porous media must be known. There is almost no resistance when loading a fibre network until a certain volume fraction is reached. At this specific volume fraction the widely separated fibres will start to make contact and the fibres will start to bend between the contact points and therefore absorb some of the load [11]. Since the number of contact points between the fibres increases must the applied force which compress the domain increase rapidly. The function for the stiffness i.e. solid pressure of the domain can be seen in Fig. 6

Figure 6: Figure over the solid pressure function.

The magnitude of the absorption is dependent of what type of fibre that is used in the fibre network [12]. The compression is described using a solid pressure term which is defined as Eq. 5.

Terazhagi’s principle states that the total applied pressure can be divided into a superposition of the hydraulic pressure and the solid pressure.

p

= p

+ p

(11)

Where p

is the solid pressure and p

is the hydraulic pressure. Even though the

total pressure is about the same as the flow direction the fibre network will deform

non uniformly. This is called stratification and it is due to that the hydraulic pressure

decreases in the flow direction and the solid pressure increases[13]. S. Toll states that

the compression of most of the fibre networks follows a power law [12], which can be

seen in Eq.12,

p

= cE(θ

− θ

). (12) C and n are material constants depending on the geometry of the fibre network. E is the elasticity modulus of the fibres and θ is the volume fraction of the fibre network, where the index 0 states the initial volume fraction of the network.

2.4 Multiphase

Multiphase simulations is used when simulating flows with different phases. Example of different phase problems can be air bubbles in water, sand particles in a fluid or even water with two different properties e.g. vapour and water in room temperature.

Multiphase is often used for problems of mixing or diffusion. Multiphase flows can be solved with two different multiphase models. First with the Eulerian-Eulerian model and second the Eulerian-Lagrange model. When simulating multiphase flows a morphology must be chosen for each phase. The different morphologies are,

•Continous Fluid.

Continous Fluid is a fluid with a continuous phase i.e. water. One example can be just ordinary flow of water in a pipe were water is a continuous phase.

•Dispersed Fluid/Solid

Dispersed Fluid/Solid is the fluid or solid which are not connected to the contin- uous fluid for example the sand when you simulate problem where you want to look at the sand particles in a fluid flow. When Dispersed fluid/solid is used a constant mean diameter must be set.

•Polydispersed Fluid/Solid

For problems where different particle or fluid sizes are important is the polydis- persed fluid/solid morphology a good option. The difference between polydispersed and dispersed fluid/solid is that the polydispersed fluid/solid particles vary in size where dispersed fluid/solid has the same size of the particles.

•Droplets.

The morphology droplets is required when problems where the droplet condensa-

tion model is important to use. One example is problems with cold water with hot

steam.

•Particle Transport Fluid/Solid.

Is the only morphology that use the Lagrangian particle tracking model instead of Eulerian-Eulerian model which are the mainly used models when modelling multi- phase flow in CFX. Lagrangian is useful when solving problems where each particle is important. Example of this is a few large particles in a fluid flow.

Multiphase flows can be solved using two options,

•homogeneous.

The homogeneous option solves the multiphase with a limited case of Eulerian- Eulerian model. Where it solves the fluids with the same velocity, pressure, turbulence and pressure fields. This option is therefore good for problems where the material data is very similar such as mixing of hot and cold water. Or a multiphase simulation of epoxy with similar properties. This saves computational resources but may not be as accurate as non homogeneous.

•non homogeneous.

The non homogeneous option solves the velocity, pressure, turbulence and pressure fields for each phase with the option of two main models mentioned earlier, Eulerian- Eulerian model and Lagrangian patricle tracking model.

2.4.1 Eulerian-Eulerian model

The Eulerian-Eulerian model describes the flow and solves the evolution of each phase for each iteration. The main equations solved in the Eulerian-Eulerian is the conti- nuity equation which can be seen in Eq. 13

δ

δt V

ρ

+ ∇ρ

V

u

= 0 (13) Index k denotes the phases. The Navier-Stokes equation can be seen in Eq. 14

δ

δt V

ρ

u

+ ∇V

ρ

u

u

= −V

∇p − ∇V

τ

± F

+ V

ρ

g (14) F denotes the interaction forces between the phases. τ denotes the viscous stress tensor. V

is the volume fraction and it is given by

V

=

P

i

N

V

V (15)

N

is the number of particles and V is the total volume of the domain. Where V

is the particle volume and it is given by Eq. 16

V

= d

i

π

6 (16)

Eulerian-Eulerian model is good for

• A wide range of volume fraction, V

.

• It is not so computational heavy.

• It includes the turbulence by default.

• The global information of the particle phase is available.

Eulerian-Eulerian model is bad for

• Many different particle sizes.

• When the knowledge of diffusion coefficient is incomplete.

• Difficult to get accuracy over a wide range of particles for combustive flow.

• When phase changes occur. Then must the new particle diameter be user specified instead of calculated.

2.4.2 Lagrangian particle tracking model

Lagrangian model solves the continuity equation and Navier-Stokes equation for the flow and the newtons second law for the particles. The continuity equation can be described as

δ

δt α

ρ

+ ∇ρ

α

u

= 0 (17) and the Navier-Stokes equation can be seen in Eq. 18

δ

δt α

ρ

u

+ ∇α

ρ

u

u

= α

∇p − ∇α

τ

+ ±F

+ α

ρ

g (18) where the index f denotes the phase of the flow and F

is the force that is acting on the flow from the particle. α

is defined in the same way as Eq. 15. The equation solved for the particle is Eq. 19.

F = m

δu

δt (19)

where u

is given from Eq. 20

u

= dx

dt (20)

Lagrangian particle tracking model is good for

• To obtain complete information about the behaviour and residence time of the particle.

• Cheaper for a wide range of particle sizes.

• Mass and heat transfer.

• For flow when the different particle sizes results in different particle veloc- ities.

Lagrangian particle tracking model bad for

• Large number of particles, it is very computational heavy.

• The model is very expensive when turbulence is required.

• It is restricted to low volume fraction.

• Only possible as a post process for a large number of particles.

2.5 Drag models

Drag models are very important when simulating multiphase flows. The models describes the friction between the both phases which is important for most of the problems. The drag coefficient, C

, can be calculated in many ways depending on the flow. For sparsely distributed particles and a low Reynolds number the drag coefficient can be described as Eq. 21.[14]

C

= 24

Re , Re << 1 (21)

As the Reynolds number increase the inertial effects overcome the viscous effect and the drag can be described as

C

= 0.44, 1000 ≤ Re ≤ 1 · 10

. (22) But for the transitional area between low Reynolds number and large raynolds numbers there is numerous models for describing the drag coefficient. The drag mod- els available in ANSYS CFX are Wen Yu model and Gidaspow for densely distributed particles.

• Wen Yu drag model is good for volume fraction up to 0.2 and probably higher than that. Wen Yu drag model equation can be seen in Eq. 23.

C

= r

24

Re

(1 + 0.15Re

) (23)

Where Re

= r

Re and r

is the continuous phase volume fraction.

• The Gidaspow model is good for very dense flows. Gidaspow drag model can be seen in Eq. 24.

C

= 150 (1 − r

)µ

r

d

+ 7

4

(1 − r

)ρ

|U

− U

| d

, r

< 0.8 (24) For r

> 0.8 the Gidaspow model follows the Wen Yu model. The index d is for the dispersed phase and d

is the particle diameter.

The drag models in the sparsely distributed particles category are Schiller Naumann, Ishii-Zuber and Grace models.

• Schiller Naumann model is good for problems where only spherical particles are used and small solid phase volume fractions.

The Schiller Naumann drag coefficient can be calculated according to Eq. 25.

C

= 24

Re (1 + 0.15Re

) (25)

• Ishii-Zuber model is good for sparsely distributed drops and bubbles. Ishii-Zuber drag coefficient is the maximum value between C

and C

.

C

is calculated in Eq. 26.

C

= 2 3 E

1 2

0

(26)

where E

is the Eotvos number which is given as, E

= gδρd

σ (27)

Eotvos number is a ratio between the gravitational and surface tension. σ is the surface tension and δρ is the density difference between the phases. But Eq. 26 is limited to a maximum value of

.

C

is defined as,

C

= 24

Re

(1 + 0.15Re

) (28) where Re

is given as

Re

= ρ|U

− U

|d

µ

(29)

µ

is defined as

µ

= µ

(1 − r

r )

(30)

r

is the user defined maximum packing value. µ

is given as, µ

= µ

+ 0.4µ

µ

+ µ

(31)

• Grace model is good for air and water systems and it is defined as,

C

= r

C

(32)

C

is the grace coefficient for a single bubble. When implementing user defined function for drag model it is important to understand how the program handle the function. The term which is implemented in the Navier-Stokes equation, Eq. 14, is given in Eq. 33.

D

= c

(U

− U

) (33)

α and β denotes the phases. The constant c

for spherical particles is defined as c

= 3

4 C

d V

ρ|U

− U

| (34)

C

is the function implemented by the user which can be seen in Eq. 35.

C

= 4µd

3V

ρ|u|k (35)

2.6 Interphase turbulence dispersion models

When simulating problems with a continuous turbulent flow with dispersed solid or fluid particles the particles can get caught up in the eddies. The only drag that is acting on the particles when they are caught is the interphase drag. To prevent this phenomenon a interphase turbulence dispersion model can be introduced. This dispersion model will result in additional dispersion of the particles from high volume fraction to low volume fraction. ANSYS have implemented two models for this, the Favre average drag model and the Lopez de Bertodano model.

2.6.1 Favre average drag model

The Favre average drag model is given by Eq. 36.

F = C

A

ν

σ

( ∇V

V

− ∇V

V

) (36)

α represents the continuous phase and β the dispersed solid or fluid. The index t

stands for turbulent, therefore is ν

the turbulent viscosity and σ

is the turbulent

Schmidt number. A

is the momentum transfer coefficient and C

is the user

modified CEL multiplier. C

is dependent on the turbulent Stokes number, St

.

Stokes number is given by Eq. 37.

St

= t

u

l

(37)

When St

= 0 is C

= 1 and when St

→ ∞ is C

= 0. t

is given by t

= ρ

d

18µ

(38)

So the Favre average drag model is only valid when m

>> m

i.e. ρ

is large and d

is small or when m

<< m

so ρ

is small. Else the t

value is too high and the resulting stokes number, St

, becomes large and the force between the phases decreases.[15, 14]

2.6.2 Lopez de Bertodano model

The equation of Lopez de Bertodando is according to, [16], given as Eq. 39

M

= −M

= C

ρ

k

∇V

(39) Where C

is a user defined constant with values ranging from 0.1 up to 500.

The momentum term is implemented into the Navier-Stokes equation directly which can be seen below,

∂u

∂t + (u · ∇)u = − 1

ρ ∇p + F + µ

ρ ∇

u + M

(40)

Extended version of the Lopez de Bertodano model is given in [17].

M

= ρ

τ

τ

( τ

τ

+ τ

)v

v

∇V

(41) τ is the turbulent Reynolds stresses and v

is the time averaged velocity fluctuation.

2.7 Solid pressure models

Solid pressure models are used to describe the particle-particle interactions in multi- phase flow so the phases don’t exceed their maximum packing fraction. There is one pre defined solid pressure model which is Gidaspow model. The Gidaspow model cal- culates the pressure difference as a function of volume fraction instead of calculating the pressure directly. Gidaspow is given in Eq. 42.

∇P

= G(v

)∇v

(42)

Where G(v

) is given in Eq. 43

G(v

) = G

e

(43)

v

is the user defined maximum packing coefficient. G

is the reference elasticity modulus. The solid pressure term is implemented directly into the Navier-Stokes equation,

∂u

∂t + (u · ∇)u = − 1

ρ ∇p + F + µ

ρ ∇

u + P

(44)

P

is one of the terms included into the outer force term F . An other way to

implement the solid pressure term into the Navier-Stokes equation is to implement it

as an elasticity modulus. This is done by defining G(v

) directly.

3 Method

3.1 Fibre Material

First the fibre material is defined in order to be able to execute the task. The material is defined as pure substance which means that the density and viscosity is defined for the material. The thermodynamic properties will be neglected because heat transfer is not of interest in this case. The density is set to ρ = 1200kg/m

.

3.1.1 Drag force

As mentioned in section 2.5 the user defined function defined as, C

in the Eq. 34.

The function is implemented according to the definition in CFX.

3.2 Geometry and mesh

The geometry with mesh used in ANSYS can be seen in Fig.7,

Figure 7: Picture over the geometry with the mesh.

Table over the properties of the geometry can be seen in the figure below,

Figure 8: Picture over the properties of the mesh and geometry.

Note that the length scale of the geometry is very small and therefore is a few number of element used. Also the one element thickness which makes the case 2- dimensional. It is not expected to happen anything in the x-direction which results in that the number of elements can be scale down in the x-direction which can be seen in Fig. 7. The small number of elements will result in a reduction of the computational resources required through the simulations. Even though there is a few number of elements is the physics captured because of the concentration of elements near the outlet.

3.3 Boundary conditions and initialization

In order to initialize the volume fraction of the fibres is the distribution showed in

Fig. 9 used. The initialization is set to the volume fraction of the fibres are 6% up to

an height of 45mm and 0% for the remaining domain. Since there is only two phases, water and fibres, is the water volume fraction is set to 94% up to 45mm and 100%

for the remaining height.

Figure 9: Figure of the initialization of the fibre volume fraction.

Pressure is set as a function of the height to help the solution to converge as the

start of the simulation, which can be seen in Fig. 10. The pressure is set so the

hydraulic pressure will drop from the static pressure set at the inlet to zero at the

bottom.

Figure 10: Figure of the initialization of the pressure drop over the domain.

The water inside of the domain is set to have a velocity of 5mm/s in the negative Y-direction. This will help the solution to convergence at the start of the simulation.

The outlet which is located in the bottom of the geometry is set to wall with a source term, Eq. 45, for the water.[18]

dM

dt = σV

F [2ρ

δP ]

. (45)

M is the mass per unit area and σ is the fractional open area of the surface. σ is set to 0.05 typically for our case. F is given in Eq. 46.

F = 1

σC − 1 (46)

C is given in Eq. 47

C = 0.6125 − 0.5148σ

1 − 0.9023σ

(47)

The boundary condition used at the inlet is an pressure driven opening with a

opening pressure of 35 kPa. The inlet is located at the top of the geometry. The

opening is set to only allow water to flow into the domain and no fibre. This will

result in that the fibres inside the domain will be driven by the water and compressed

against the outlet. The front and back of the geometry is set to symmetry so be able

to simulate a 2-d problem. The left and right side of the geometry is set to wall with

free slip condition. The free slip condition says that the flow in the normal direction to

the wall will be zero and it is only calculating the velocity gradient parallel to the wall.

4 Results

Contour plots over the results when steady state solution is reached for pulp 1 can be seen in Fig. 11. The plot to the left represents the hydraulic pressure, the plot in the center is the volume fraction and the one to the right is the solid pressure.

Figure 11: Contour plot over the hydraulic pressure, solid pressure and volume frac-

tion fibre for pulp 1 when steady state solution is reached.

Figure 12: Contour plot over the hydraulic pressure, solid pressure and volume frac- tion fibre for pulp 2 when steady state solution is reached.

The left contour is the hydraulic pressure, center contour is volume fraction of fibre and the contour to the right is the solid pressure. In Fig. 12 can the result of pulp 2 be seen. The pulp has solid pressure function which is less stiff than for pulp 1. It is clear if a comparison of the volume fraction distribution through the domain is done between Fig. 11 and Fig. 12. Fibres in pulp 2 is more compressed and the pressure contours are a bit different as a result.

In Fig. 13 the figure over volume fraction for two different pulps can be seen.

The red line represents pulp 2 and the blue line represents pulp 1. The x-axis is the

volume fraction and the y-axis is the length of the domain where 0 is the area closest

to the outlet.

Figure 13: Picture over the volume fractions for pulp 1 and pulp 2

Figure over Terazaghi’s principle can be seen in Fig. 14. It’s clear that the principle

is full filled for both of the pulps. The blue lines represents pulp 1 and the red lines

represents pulp 2. The darker colour represents the hydraulic pressure and the less

dark ones represents the solid pressure.

Figure 14: Picture over Terazaghi’s principle for both of the pulps.

5 Discussion and Conclusion

Because of the sensitivity of the permeability and solid pressure the numerical solver is unable to stabilize the solution and the numerical solver therefore needs modifi- cations in order to be able to execute the simulations. The results obtained seems accurate even though the solver still has some minor instability. It’s also shown that it’s very important that the data for the pulps is accurate and valid, else the solver will crash or the solution is wrong. In section, 4, it can see how a small difference in solid pressure will effect the solution. Pulp 1 has a stiffer solid pressure function than pulp 2. Too large modification will result in that the solver crashes due to that the pressure gradient goes to infinity as the volume fraction of the fibres goes to V

= 1.

Results for the both pulps seems accurate and it’s known that a maximum volume fraction of V

= 0.15 is realistic. The hydraulic and solid pressure is accurate too.

In Fig. 14 it can also be seen that Therazhagi’s principle is full filled for both cases.

The sum of the solid pressure and the hydraulic pressure is equal to the static inlet pressure of 35kPa, which is expected. The volume fraction graphs seen in Fig. 13 looks realistic as well. The difference of the pulps comes from the different param- eters in the solid pressure function and it’s obvious that pulp 1 has a stiffer solid pressure equation and therefore is harder to compress.

Conclusions which can be drawn is that there is limitations in the standard solver

which can be manage by changing the solver strategy by using different kinds of

relaxation factors and interpolation methods. The solver is also sensitive when it

comes to solid pressure and it’s therefore important that the constants are correct for

the solid pressure equation.

6 Future work

The method of deriving the constants for solid pressure and permeability should be investigated. Since very small changes in these parameters will effect the solution, hence the values for each pulp and the method for how they’re obtained needs to be correct. The solid pressure function can be modified in order to capture the effects which occur when the compression velocity is high. One effect which occur is the phenomenon of micro channels. Instead of filtrating evenly between the fibres micro channels are created where the fluid is channelled instead which completely changes the permeability. Consequently the fibre network is unevenly distributed across the surface. This effect may be modified by a velocity dependent term in the solid pres- sure function. Schematically picture over micro channel can be seen in Fig. 15.

Figure 15: Figure over the channels created when filtration rate is too high. Source:

[4]

Another effect is that the dewatering rate of the water inside of the fibres is lim- ited. Therefore will the fibres be more stiff if the compression velocity is high. This will result in that the fibre suspension will act more stiff if the compression velocity is high. So the current solid pressure function will underestimate the solid pressure if the velocity is too high. This will cause a collapsing phenomenon where the perme- ability goes to zero and the pressure gradient to infinity.

Another work can be to implement fibres of different sizes. The implementation

can be done with the morphology polydispersed solid instead of dispersed solid which

is used in this case. Test of different numerical solvers can also be one case, where

the numerics is compared to find the most stably one.

7 Appendix References

[1] Persson Knut-Erik. PAPPERSTILLVERKNING. ISBN 91-7322-190-2.

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[12] S Toll. Packing mechanics of fiber reinforcements. Polymer Engineering and

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[14] Ansys CFX User Guide version 16.2 Februrary 2016.

[15] Th Frank, P. J. Zwart, E. Krepper, H. M. Prasser, and D. Lucas. Validation of CFD models for mono- and polydisperse air-water two-phase flows in pipes.

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