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Modelling of heat and moisture transport

in a corrugated board stack

MARIANNA XYNOU

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ABSTRACT

The corrugated board is considered as the second most used packaging material and the world’s environmentally acceptable solution for packaging, with wide range of applications. After the manufacturing process, the corrugated board is cut into sheets and stored in a stack until optimum moisture content has been reached in order to avoid undesired properties.

However, due to complex and various structures, it is difficult to estimate the appropriate time so to achieve the acceptable moisture level of the corrugated board stack. So a homogenized model of the stack has to be created which will have the same average properties as the real stack. In order to achieve this goal the behavior of a smaller part of the stack, the unit cell, is investigated. In the second step a homogenized model is created with the average transport of mass and heat. At the end, the unit cell is scaled up.

In this master thesis, only the first and the second steps were simulated. This was achieved by creating a 3-D mathematical model using finite element method and simulating its properties in COMSOL Multiphysics®. Four mathematical models were used in the description of the 3-D model: the heat transfer, the moisture transfer, the vapour concentration and the gas pressure. Moreover, by applying the gradient in one direction in each case, the behavior of the detailed unit cell was investigated. Finally different simplified geometries were created and investigated so to approach a homogenized model which described better the average properties of the detailed model.

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Acknowledgements

I have been fortunate to cooperate with many people who have given me assistance, knowledge and inspiration for completing this thesis.

First of all, I would like to thank my supervisor Jan-Erik Gustafsson, and also Johan Alfthan for the chance which gave me and also to work in a creative and friendly environment. Moreover I am grateful to SUW (The International Development Group for Corrugated Board), for the opportunity to do this masters thesis. It has been an interesting challenge for me to meet and talk with the members of SUW.

My kindest acknowledgments are made to Matthäus Bäbbler for helping me with his knowledge and that he always pushes me to think further.

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Content

List of Tables ... 4 List of Figures ... 5 Nomenclature ... 8 1. Introduction ... 1 1.1. History ... 1

1.2. Structure of Corrugated board ... 1

1.3. Terminology ... 2

1.4. Manufacturing process ... 3

2. Aim of the project ... 4

3. Transport Phenomena ... 5

Conduction ... 5

Convection... 5

Diffusion ... 5

4. Mathematical models... 6

Heat transfer model ... 6

Moisture model ... 6

Vapour model ... 7

Gas pressure model –Brinkman equation ... 7

5. Modelling ... 8

5.1. Boundary conditions ... 8

5.1.1. Detailed cell ... 8

5.1.2. Homogenized models ... 11

Homogenized model 1- z direction ... 11

Homogenized model 2- z direction ... 12

Homogenized model 3- z direction ... 13

6. Simulation of unit cell of the corrugated board ... 15

7. Parameters ... 16

8. Results - Discussion ... 17

8.1. Detailed model 1... 17

8.1.1. Case 1a: Temperature gradient in the direction ... 17

8.1.2. Case 1b: Temperature gradient in x the direction ... 20

8.1.3. Case 1c: Temperature gradient in y the direction ... 21

8.1.4. Case 2a: Moisture gradient in z the direction ... 22

8.1.5. Case 2b: Moisture gradient in x the direction ... 23

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8.1.7. Case 3a: Vapour concentration gradient in z the direction ... 26

8.1.8. Case 3b: Vapour concentration gradient in x the direction ... 27

8.1.9. Case 3c: Vapour concentration gradient in y the direction ... 28

8.1.10. Case 4a: Pressure gradient in z the direction ... 30

8.1.11. Case 4b: Pressure gradient in x the direction... 31

8.1.12. Case 4c: Pressure gradient in y the direction ... 32

8.1.13. Flow balance and accumulation ... 33

8.1.14. Transport coefficients ... 33

8.2. Detailed model 2... 35

8.2.1. Case 1a: Temperature gradient in z the direction ... 35

8.3. Homogenized model 1 ... 37 8.4. Homogenized model 2 ... 38 8.5. Homogenized model 3 ... 39 9. Conclusions ... 40 Bibliography ... 41 Appendix A ... 42 Appendix B... 50 Appendix C... 51 Appendix D ... 58 Appendix E ... 72 Appendix F ... 83

List of Tables

Table 1: Different types of corrugated board [2] ... 2

Table 2: Table 2: Initial values of the cell ... 8

Table 3: Matrix for COMSOL Multiphysics® simulation ... 9

Table 4: Dimension of the detailed model... 15

Table 5: Parameters of the detailed model ... 16

Table 6: Values of the heat fluxes for different temperature gradients ... 17

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List of Figures

Figure 1 : Single face corrugated board ... 1

Figure 2: Different types of single and double corrugated board ... 1

Figure 3: Characteristics properties of the corrugated board... 2

Figure 4: Cell of the corrugated board ... 2

Figure 5: Corrugated board manufacturing process [5] ... 3

Figure 6: Single and double facer corrugated board ... 3

Figure 7: Names of the cell's surfaces ... 8

Figure 8: Geometry of the homogenized model 1 ... 11

Figure 9: Geometry of the homogenized model 2 ... 13

Figure 10: Geometry of the homogenized model 3 ... 13

Figure 11: Detailed cell with the dimension ... 15

Figure 12: Mesh of the detailed model ... 15

Figure 13: Total heat flux as a function of the temperature gradient applied in the z direction ... 17

Figure 14: Vector plot of the total heat flux for y=5mm. ... 17

Figure 15: Total moisture flux as a function of the temperature gradient applied in the z direction ... 18

Figure 16: Vector plot of the total moisture flux for y=5mm ... 18

Figure 17: Total vapour flux as a function of the temperature gradient applied in the z direction ... 18

Figure 18: Vector plot of the total vapour flux for y=5mm... 18

Figure 19: Velocity vector as a function of the temperature ... 18

Figure 20: Temperature behaviour in surface z=3.944mm ... 19

Figure 21: Total heat flux (Qz) behaviour in surface z=3.944mm ... 19

Figure 22: Total heat flux as a function of the temperature gradient applied in the x direction ... 20

Figure 23: Total moisture flux as a function of the temperature gradient in x direction ... 20

Figure 24: Total vapour flux as a function of the temperature gradient applied in the x direction ... 20

Figure 25: Velocity vector as a function of the temperature gradient applied in the x direction ... 20

Figure 26: Total heat flux as a function of the temperature gradient applied in the y direction ... 21

Figure 27: Total moisture flux as a function of the temperature gradient applied in the y direction ... 21

Figure 28: Total vapour flux as a function of the temperature gradient applied in the y direction ... 21

Figure 29: Velocity vector as a function of the temperature gradient applied in the y direction ... 21

Figure 30: Total heat flux as a function of the moisture gradient applied in the z direction ... 22

Figure 31: Total moisture flux as a function of the moisture gradient applied in the z direction ... 22

Figure 32: Total vapour flux as a function of the moisture gradient applied in the z direction ... 22

Figure 33: Velocity vector as a function of the moisture gradient applied in the z direction .. 22

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Figure 35: Total moisture flux as a function of the moisture gradient in the x direction ... 23 Figure 36: Total vapour flux as a function of the moisture gradient applied in the z direction ... 23 Figure 37: Velocity vector as a function of the moisture gradient applied in the x direction . 23 Figure 38: Total heat flux as a function of the moisture gradient applied in the y direction... 24 Figure 39: Total moisture flux as a function of the moisture gradient applied in the y direction ... 24 Figure 40: Total vapour flux as a function of the moisture gradient applied in the y direction ... 24 Figure 41: Velocity vector as a function of the moisture gradient applied in the y direction . 24 Figure 42: Moisture flux for the moisture gradient in the y direction equal to 0.006, for

x=3.885mm... 25 Figure 43: Vapour flux for the moisture gradient in y direction equal to 0.006, for x=3.885mm ... 25 Figure 44: Total heat flux as a function of the vapour concentration gradient applied in the z direction ... 26 Figure 45: Total moisture flux as a function of the vapour concentration gradient applied in the z direction ... 26 Figure 46: Total vapour flux as a function of the vapour concentration gradient applied in the z direction ... 26 Figure 47: Total velocity vector as a function of the vapour concentration gradient applied in the z direcdtion ... 26 Figure 48: Total heat flux as a function of the vapour concentration gradient applied in the x direction ... 27 Figure 49: Total moisture flux as a function of the vapour concentration gradient applied in the x direction ... 27 Figure 50: Total vapour flux as a function of the vapour concentration gradient applied in the x direction ... 27 Figure 51: Total velocity vector as a function of the vapour concentration gradient applied in the x direction ... 27 Figure 52: : Total heat flux as a function of the vapour concentration gradient applied in the y direction ... 28 Figure 53: Total moisture flux as a function of the vapour concentration gradient applied in the y direction ... 28 Figure 54: Total vapour flux as a function of the vapour concentration gradient applied in the y direction ... 28 Figure 55: Total velocity vector as a function of the vapour concentration gradient applied in the y direction ... 28 Figure 56: Vapour flux, for vapour concentration gradient 0.0006 kg/m3 in z the direction . 29 Figure 57: Vapour flux, for vapour concentration gradient 0.001 kg/m3 in z the direction .... 29 Figure 58: Total heat flux as a function of the pressure gradient applied in the z direction ... 30 Figure 59: : Total moisture flux as a function of the pressure gradient applied in the z

direction ... 30 Figure 60: Total vapour flux as a function of the pressure gradient applied in the z direction ... 30 Figure 61: Total velocity vector as a function of the pressure gradient applied in the z

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Figure 63: : Total moisture flux as a function of the pressure gradient applied in the x

direction ... 31

Figure 64: Total vapour flux as a function of the pressure gradient applied in the x direction 31 Figure 65: Total velocity vector as a function of the pressure gradient applied in the x direction ... 31

Figure 66: Total heat flux as a function of the pressure gradient applied in the y direction ... 32

Figure 67: Total moisture flux as a function of the pressure gradient applied in the y direction ... 32

Figure 68: Total vapour flux as a function of the pressure gradient applied in the y direction 32 Figure 69: : Total velocity vector as a function of the pressure gradient applied in the y direction ... 32

Figure 70: Total heat flux as a function of the temperature gradient applied in the z direction ... 35

Figure 71: Total moisture flux as a function of the temperature gradient applied in the z direction ... 35

Figure 72 :Total vapour flux as a function of the temperature gradient applied in the z direction ... 35

Figure 73: Velocity vector as a function of the temperature gradient applied in the z direction ... 35

Figure 74: Detailed model 1: vector plot of the total heat flux for y=5mm. ... 36

Figure 75: Detailed model 2: Vector plot of the total heat flux for y=5mm. ... 36

Figure 76: Temperature behaviour in surface z=3.944mm ... 36

Figure 77: Total heat flux as a function of the temperature gradient applied in the z direction ... 37

Figure 78: Total moisture flux as a function of the temperature gradient applied in the z direction ... 37

Figure 79: Total vapour flux as a function of the temperature gradient applied in the z direction ... 37

Figure 80: Velocity vector as a function of the temperature gradient applied in the z direction ... 37

Figure 81: Total heat flux as a function of the temperature gradient applied in the z direction ... 38

Figure 82: Total moisture flux as a function of the temperature gradient applied in the z direction ... 38

Figure 83: Total vapour flux as a function of the temperature gradient applied in the z direction ... 38

Figure 84: Velocity vector as a function of the temperature gradient applied in the z direction ... 38

Figure 85: Total heat flux as a function of the temperature gradient applied in the z direction ... 39

Figure 86: Total moisture flux as a function of the temperature gradient applied in the z direction ... 39

Figure 87: Total vapour flux as a function of the temperature gradient applied in the z direction ... 39

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Nomenclature

Cpa specific heat capacity of the dry air, J/(kg∙K) Cpftot total specific heat capacity of the flutes J/kg∙K) Cpgtot total specific heat capacity of the glue J/kg∙K) Cpl specific heat capacity of the liquid water, J/(kg∙K) Cpptot total specific heat capacity of the paper J/kg∙K) Cps specific heat capacity of the dry fibre, J/(kg∙K)

Cpsg specific heat capacity of the dry solid in glue, J/(kg∙K) Cpv specific heat capacity of the vapour, J/(kg∙K)

Cvp0 initial vapour concentration in the air inside pores, kg/m3 Cvp vapour concentration in the air inside pores, kg/m3

D diffusion coefficient for liquid water in fibre network, m /s2 DL diffusion coefficient for liquid water in glue, m /s2

Dv diffusion coefficient for vapour in air, m /s2

Dv352K gas diffusion coefficient of vapour in air at 352 K, m /s2 eps porosity of the paper

R evaporation rate 1/s

Fr Sourse term evaporation rate 1/s gR gas constant, J/(mol K)

hita dynamic viscosity of the liquid Pa∙s Kgpa mass transfer coefficient, s-1

kkapa hydraulic permeability m2

lamdaa thermal conductivity of the air W/(m∙K) lamdaftot thermal conductivity of the glue W/(m∙K) lamdagtot total thermal conductivity of the glue W/(m∙K) lamdal thermal conductivity of liquid air W/(m∙K) lamdaptot thermal conductivity of the glue W/(m∙K) lamdas thermal conductivity of dry fibre W/(m∙K)

lamdasg thermal conductivity of dry solid in glue W/(m∙K) lamdasorp Specific heat of sorption J/K

lamdaT Specific heat of evaporation of water J/K lamdav thermal conductivity of of the vapour W/(m∙K) Ma molar mass of the air kg/mol

Mw molar mass of water, kg/mol p0 initial gas pressure, Pa pa partial pressure of the air, Pa

pp partial pressure of the water in the paper, Pa psat saturation pressure for the free water, Pa

psat0 initial saturation pressure in the paper, glue and air, Pa ptot normal atmospheric pressure, Pa

pv 296K partial pressure of vapour in the surrounding air at = temperature 296K, Pa pv partial pressure of the vapour in the pores, Pa

phi relative humidity Qhs heat source, W/m3 rhoa density of the air, kg/m3

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rhop density of the paper kg/ m3 rhoptot total density of the paper kg/m3 rhosg density of dry solid in glue kg/m3 rhos density of the dry fibre kg/m3 ssigma_m mass transfer coefficient kg/s∙m2

T0 initial temperature of the paper board and glue, K u0 initial moisture content in the paper, glue and air, ratio u moisture content calculated from the moisture model, ratio ufsp moisture content at saturation point

v velocity of the fibre network relative to the coordinate system, m/s

Greek letters

ε porosity of the paper

η dynamic viscosity of the fluid, kg/(m∙s)

κ hydraulic permeability- depends on porosity, m2 φrelative humidity

ρ density, kg/m3

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1. Introduction

1.1. History

The history of corrugated board starts 150 years ago in 1856, when pleated paper was patented in England and used as a liner for tall hats and ruffled Elizabethan collars. In 1871 ‘the father of the corrugated board’, American Albert L. Jones, created the first patent issued for corrugated paper as a packaging material and designed the single face corrugated board (Figure 1) [1].

Figure 1 : Single face corrugated board

Since then, many changes and progress have taken place in the improvement of the material and the industrial equipment. Over the years the interest of the customers has increased so that the requirements towards corrugated board enlarged and nowadays different types exist (Figure 2).

Figure 2: Different types of single and double corrugated board

Nowadays, corrugated board plays an important role in the packaging industry. About 30% of packaging today is made from based packaging materials and paper-based materials are the second most used packaging after plastics.

1.2. Structure of Corrugated board

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can have different types, with their characteristics being the height and the pitch as described in Figure 3. Moreover in Table 1 are presented the dimensions of different types that exist, which are specified according to flute height and pitch [2].

Figure 3: Characteristics properties of the corrugated board Table 1: Different types of corrugated board [2]

Types Flute height

(mm) Flute Pitch (mm) A 4.0-4.8 8.0-9.5 B 2.1-3.0 5.5-6.5 C 3.2-3.9 6.8-7.9 E 1.0-1.8 3.0-3.5 F 0.75 1.9-2.6 G 0.55 <1.8 1.3. Terminology

Corrugated medium or fluting: a combination of hardwood-fibres and corrugated containers, from chemical and semi-chemical pulp. Typically the basic weight of the medium varies from 112 to 195 g/m2 with target moisture of 7-9 % [3], [4].

Liner: is usually made of softwood kraft pulp, waste corrugated container or a combination of both. The average weight varies from 112 to 439 g/m2 with moisture of 4-8 % [3].

Glue: is commonly made from different types of starch [3].

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3 1.4. Manufacturing process

As it is observed in Figure 5, the manufacturing process of the corrugated board consists of mainly two steps, the single and the double facer operation.

Figure 5: Corrugated board manufacturing process [5]

There is single and double facer corrugated board (Figure 6). On this paragraph the manufacturing operations are mentioned briefly [3], [5].

 Single facer operation

During this process a liner and a fluted medium are glued together at the flute tips. In this part of the operation three different steps are taken place:

1. Unwinding and tension control 2. Preheating and preconditioning 3. Flute forming and bonding  Double facer operation

In the second part of the process the single faced board is joined with a liner in order to achieve the final design, while the process occurs in two sections :

1. Heating section 2. Traction section

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2. Aim of the project

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3. Transport Phenomena

In this section the phenomena which occur due to the moisture and heat transport are presented, when the board is cut and stored in a stack of corrugated board until suitable moisture content is reached.

Conduction

Conduction describes the transfer of heat through a medium when there is temperature gradient. The effectiveness of the heat transfer is measured through thermal conductivity, λ, where high conductivity indicates a good conductor.

Furthermore, the conductivity of the board depends on the moisture content, and the thermal conductivity increases by increasing moisture content and temperature [6]. Convection

This phenomenon affects both heat and moisture transfers in the corrugated board [7]. Convection is the transfer of heat by the actual movement of the warmed material and occurs in liquids and gases, when pressure gradient exists. The mechanism is based on the movement of the vapour to colder area where it can condensate and the latent heat of the vaporization is released. This releases the latent heat of vaporization in the area of condensation and can increase the temperature in that area [6].

Diffusion

Diffusion is the transport of molecules from an area with higher concentration to an area with lower concentration, by random molecular motion.

Liquid diffusion coefficient: is a function of the moisture content and increases exponentially with moisture content above the fibre saturation point, while it has linear relationship with the temperature [6].

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4. Mathematical models

The modelling of heat and moisture transport for the 3-D geometry is complicated because of the complex structure of the corrugated board. In order to simplify the model some assumptions are taken into consideration, such as:

 The permeability is constant and considered the same for the paper and the glue.

 The dynamic viscosity is assumed to be constant.  The liquid diffusion coefficient is constant in the paper.  Local thermal equilibrium exists.

 Vapour and air transport are described by diffusion and convection phenomena.

 The heat transport is described by the convection and conduction phenomena.  The phenomenon of shrinkage of the paper is not taken into account

Heat transfer model

This model calculates the temperature distribution of the corrugated board that is governed by conduction and convection.

( ) ,

With:

T: Temperature [K],

: Wet density of the paper

: Total heat capacity of the paper

v: Gas velocity vector

: Total thermal conductivity of the paper

: Heat source available from specific heat of vaporization of water and specific heat of sorption .

For calculating the heat transfer in the glue and the flutes the parameters are formed according to the material properties (Appendix A).

Moisture model

The moisture transport is described through the moisture distribution in the corrugated board through convection and diffusion phenomena.

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7 With:

u: Moisture

: Diffusion coefficient for liquid water in the paper

: Source term of mass creation available from the evaporation rate of moisture in the paper and glue [s-1]

For calculating the heat transfer in the glue and the flutes the parameters are adjusted according to the material (Appendix A).

Vapour model

Even when the stack of the corrugated boards is stored, it has higher temperature comparing with the temperature of the environment so evaporation of the water occurs and moves through the fibres. The release of the water is described through diffusion and convection model(see Appendix A).

( [ ])

( [ ]) [ ],

With:

: Porosity of the paper (volume of the pore over the total volume = constant) [Cvp] : Concentration of vapour in the pores of the paper or vapour density

: Diffusion coefficient for water vapour in the air pores .

Gas pressure model –Brinkman equation

In order to express the flow in the porous media (paper) two models can be used, Brinkman equation and Darcy’s law. In this study the Brinkman equation is selected to simulate the gas pressure of the corrugated board. The following equation combines the porous media and the free flow domains for the flutes (Appendix A).

( ) ( ) [ ( ) { ( ( ) ) ( )}], With:

: Density of the gas : Density of the paper

: Dynamic viscosity of the liquid

: Hydraulic permeability (depends upon porosity of medium)

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5. Modelling

5.1. Boundary conditions 5.1.1. Detailed cell

In the unit cell the external boundaries have important role in the simulation of the model. In order to describe the boundaries conditions on the surfaces of the cell, the walls are named as specified in Figure 7:

Figure 7: Names of the cell's surfaces

The unit cell has been selected to be located in the center of the stack. It is chosen the gradient to apply for one variable in every case. Furthermore, the initial values have been selected in order the unit cell to be in steady state before any disturbance is applied. More specific, for quantity X which stands for T, u, Cvp or p, gradient ΔX is

applied in the unit cell where it sets X2>X1 on diametrically opposite surface, with

X2= X+ΔX/2 and X1= X-ΔX/2. The fluxes are calculated in different positions on five

cut planes in each direction (see Appendix B).

The initial values of the unit cell are presented in Table 2. Table 2: Table 2: Initial values of the cell

Variables Symbols Values

Temperature T0 303K = 30 oC Moisture uo 0.08053 Vapour concentration Cvp0 0.014 kg/m3 Pressure p0 1.013 105 Pa

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Table 3: Matrix for COMSOL Multiphysics® simulation

Case a b c Variable\ direction z x y 1 T 2 u 3 Cvp 4 p

Two different detailed models are investigated with dissimilar boundary conditions. In the first model, the values of the parameters in the six surfaces are fixed. Furthermore, in each case the gradient is applied only in one direction while in all the surfaces there is flux. While in the second model, the gradient will appear in one direction, as in the first one, though the fluxes will exist only in the same direction as the gradient and the other four surfaces will be insulated.

More specific, the boundary conditions for the detailed model 1 which are simulated with COMSOL Multiphysics® are presented in the four different cases:

Case 1: Temperature gradient in z-direction

z2 surface BC No Flux Fixed T - T2 u - u0 Cvp - Cvp0 p - p0 z1 surface BC No Flux Fixed T - T1 u - u0 Cvp - Cvp0 p - p0 x2, x1, y2, y1 surface BC No Flux Fixed T - T0 u - u0 Cvp - Cvp0 p - p0

Case 2: Moisture gradient in z-direction

z2 surface BC No Flux Fixed T - T0 u - u2 Cvp - Cvp0 p - p0 z1 surface BC No Flux Fixed T - T0 u - u1 Cvp - Cvp0 p - p0 x2, x1, y2, y1 surface BC No Flux Fixed T - T0 u - u0 Cvp - Cvp0 p - p0

Case 3: Vapour concentration gradientin z-direction z2 surface BC No Flux Fixed T - T0 u - u0 Cvp - Cvp2 p - p0 z1 surface BC No Flux Fixed T - T0 u - u0 Cvp - Cvp1 p - p0 x2, x1, y2, y1 surface BC No Flux Fixed T - T0 u - u0 Cvp - Cvp0 p - p0

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The same boundaries conditions exist when the gradient is applied in the x and the y direction. This results in 12 cases that are investigated for this model.

Similarly the boundary conditions of the four cases are presented for the detailed model 2:

Case 1: Temperature gradient in z-direction

z2 surface BC No Flux Fixed T - T2 u - u0 Cvp - Cvp0 p - p0 z1 surface BC No Flux Fixed T - T1 u - u0 Cvp - Cvp0 p - p0 x2, x1, y2, y1 surface BC No Flux Fixed T + - u + - Cvp + - p + -

Case 2: Moisture gradient in z-direction

z2 surface BC No Flux Fixed T - T0 u - u2 Cvp - Cvp0 p - p0 z1 surface BC No Flux Fixed T - T0 u - u1 Cvp - Cvp0 p - p0 x2, x1, y2, y1 surface BC No Flux Fixed T + - u + - Cvp + - p + -

Case 3: Vapour concentration gradientin z-direction z2 surface BC No Flux Fixed T - T0 u - u0 Cvp - Cvp2 p - p0 z1 surface BC No Flux Fixed T - T0 u - u0 Cvp - Cvp1 p - p0 x2, x1, y2, y1 surface BC No Flux Fixed T + - u + - Cvp + - p + -

Case 4: Pressure gradientin z-direction z2 surface BC No Flux Fixed T - T0 u - u0 Cvp - Cvp0 p - p2 z1 surface BC No Flux Fixed T - T0 u - u0 Cvp - Cvp0 p - p1 x2, x1, y2, y1 surface BC No Flux Fixed T + - u + - Cvp + - p + -

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The purpose of the homogenization is to simplify the detailed cell model of the corrugated board in order for all the types of the corrugated board the calculations to be less complicated for the moisture and the heat transport. The geometry of the detailed cell is simplified and simulated as a homogeneous unit cell which consist of a new material. This material will have the average properties of the paper, the glue and the air according to their mass or volume proportion or by the resistance that the new model will have. The equations that are going to describe the new properties of the homogenized unit cell are according to its geometry.

As the cell does not have the same geometry on its different axis, the equations that describe the homogenized model will be different on each one of the three directions so the homogenization process has to take place only on one direction each time. In this thesis, the homogenization in the z axis of the cell is simulated. As a result, the gradient is applied only in the z direction and the fluxes will exist in the same direction, while the other four surfaces, of the x and y direction of the cell, will be insulated. So there will be the same boundary conditions as in the second detailed model.

Homogenized model 1- z direction

The first model which is built has simple geometry, where the glue is placed as a thin layer on the top of the bottom liner and the fluting is straightened and placed in the middle of the cell, in parallel with the liners (Figure 8). An important factor of the new homogenized geometry is that the mass of the materials (paper, glue and air) will be the same in both the detailed and the homogenized cell of the corrugated board.

Figure 8: Geometry of the homogenized model 1

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12 Thermal conductivity:

Eq: 1, Diffusion coefficient for liquid: ( ) ( ) Eq: 2,

Diffusion coefficient for vapour:

Eq: 3, where : the total height of the unit cell.

Moreover, the variables of evaporation rate, the heat source, the mass source term, the specific thermal capacity are calculated according to the mass of the paper, the glue and the flutes and density according to the volume of them, as it is showed below: Evaporation rate: ( ) Eq: 4 Source term: ( ) Eq: 5 Heat source: ( ) Eq: 6

Specific thermal capacity: ( ) ( ) ( ) Eq: 7 Density: ( ) ( ) ( ) Eq: 8

Finally the moisture in the detailed models is present only in the mass of the paper and the glue. As a result, the initial value of the moisture will change as is described above:

Moisture: (

) Eq: 9

Homogenized model 2- z direction

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Figure 9: Geometry of the homogenized model 2 Thermal conductivity:

(

( ) ( ) ( ) )

Eq: 10, Diffusion coefficient for liquid:

( ) ( ) Eq: 11,

Diffusion coefficient for vapour:

(

)

Eq: 12, where : the total height of the unit cell.

The equations which are based on the mass and the volume of paper, glue and flutes are going to be the same as the homogenized model 1, as are presented in the equations Eq: 4- Eq: 9.

Homogenized model 3- z direction

The third homogenized model is based on the fact that on the detailed cell the fluting is connected with the top and the bottom liner but also with the surfaces in the x direction. So the fluting can exist as a cross in the centre of the cell, while the liners and the glue appear with the same geometry as in the previous two models. As it is showed in Figure 10, the homogenized model is a combination of the two previous homogenized models.

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According to this design the resistance is in parallel and in series so the thermal conductivity, the vapour and the liquid diffusion coefficient are calculated as presented in the equations Eq: 13- Eq: 15 and the calculation of the equations are explained in Appendix C: Thermal conductivity: ( ( )( ) ( ) ( ) ) Eq: 13,

Diffusion coefficient for liquid:

( ) ( ) Eq: 14,

Diffusion coefficient for vapour:

( ( )( ) ( ( ) ( )) ) Eq: 15,

where : the total height of the unit cell.

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6. Simulation of unit cell of the corrugated board

The dimensions of the unit cell have been selected according to ISO standards for the flute pitch and height which were presented in Table 1. The characteristics of the unit cell are presented in Table 4 and the dimensions are illustrated in Figure 11.

Table 4: Dimension of the detailed model

Cell dimensions (mm) L1 0.269 L2 0.264 L3 0.263 L4 3.411 L5 7.77 L6 10

Figure 11: Detailed cell with the dimension

The results have been calculated for time 3000s when the model reaches steady state. In Figure 12 is presented the mesh which the detailed models are simulated in COMSOL Multiphysics®, which consists of 16346 domain elements, 6422 boundary elements and 936 edge elements.

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7. Parameters

Table 5: Parameters of the detailed model

COMSOL symbol

Mathematical symbol

Value Description Reference

Cvp0 Cvp0 0.014[kg/m^3] Initial vapour concentration selected

Cp_s Cps 1300[J/(kg∙K)] Specific heat capacity of dry fibre [6]

Cp_l Cpl 4190[J/(kg∙K)] Specific heat capacity of liquid water [6]

Cp_v Cpv 1870[J/(kg∙K)] Specific heat capacity of vapour [6]

Cp_sg Cpsg 1202[J/(kg∙K)] Specific heat capacity of glue dry solid [8]

Cp_a Cpa 1005[J/(kg∙K)] Specific heat capacity of dry air [6]

D D 10^-9[m2∙s] Diffusion coefficient for paper [6]

D_v352 Dv352 3.57∙10-5[m2/s] Diffusion coefficient of vapour in air at 352K

[6]

eps ε 0.5 Porosity of the paper [6]

h_ita η 1.86∙10-5[Pa∙s] Dynamic viscosity of the liquid [9]

kg_pa kgpa 100[1/s] Mass transfer coefficient [9]

k_kapa κ 10-14[m2] Hydraulic permeability [9]

lamda_a λa 0.026[W/(m∙K)] Thermal conductivity of the air [6]

lamda_l λl 0.64[W/(m∙K)] Thermal conductivity of liquid air [6]

lamda_s λs 0.157[W/(m∙K)] Thermal conductivity of dry fibre [6]

lamda_sg λsg 0.216[W/(m∙K)] Thermal conductivity of dry solid in glue

[10]

lamda_v λv 0.019[W/(m∙K)] Thermal conductivity of of the vapour [9]

M_a Ma 28.8∙10-3[kg/mol] Molar mass of the air [6]

M_w Mw 18[kg/kmol] Molar mass of water [6]

p0 p0 1.013∙105[Pa] Initial pressure selected

P_296 P296 1450[Pa] Partial pressure of vapour in the surrounding air at 296K [6] rho_l ρl 998[kg/m 3 ] Density of water [9] rho_p ρp 780[kg/m 3

] Density of the paper [9]

rho_sg ρsg 1550[kg/m

3

] Density of dry solid in glue [9]

rho_s ρs 1560[kg/m

3

] Density of of the dry fibre [9]

R_g Rg 8.314[J/(mol∙K)] Ggas constant [6]

s_sigma_m σ 0.055[kg/s∙m2] Mass transfer coefficient [9]

T0 T0 303 [K] Initial temperature selected

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17

8. Results - Discussion

8.1. Detailed model 1

In this chapter the results of the 12 cases are presented and analysed through the fluxes for the detailed model 1.

As a general rule the sign of the fluxes are determined with reference to the underlying vector field, so a flux is positive if it goes in positive direction.

8.1.1. Case 1a: Temperature gradient in the direction

On this case temperature gradient is applied only in the z direction, where T2 is the

temperature on the top surface and T1 on the bottom surface and T2> T1.

Figure 13: Total heat flux as a function of the temperature

gradient applied in the z direction Figure 14: Vector plot of the total heat flux for y=5mm. As it is showed in Figure 13, the temperature gradient in z direction causes fluxes of all quantities to all the surfaces of the detailed cell.

As it is expected the heat flux in z direction (Qz) has linear behaviour as the

temperature gradient increases. More conclusions can be drawn by observing Figure 14, which shows the cut plane that visualises the vector filed of the total heat flux in arrows on a 2D surface and y=5mm. It is observed that in the flutes, the heat moves in and out of the cell; so that the lines of the heat flux inside the cell do not have the same slope with those of the bottom and the top surfaces. After this observation, it is interesting to calculate the values of the cross terms (Qx and Qy) (see Table 6). The

values of the Qx and Qy on the surfaces z=0 and z=3.944mm are comparable and the

difference can be explained as phase change occurs.

Table 6: Values of the heat fluxes for different temperature gradients

Total heat flux, Qz

(W/m^2)

Total heat flux, Qx

(W/m^2)

Total heat flux, Qy

(W/m^2)

z\DT 0 4 6 x\DT 0 4 6 y\DT 0 4 6

0 0.021 -89.01 -178 0 0.0056 23.88 47.79 0 0.0047 -7.304 -14.55

0.986 -7E-04 -16.76 -33.5 1.943 4.2E-05 -2.95 -5.881 2.5 2.76E-04 0.182 0.369

1.972 0.00007 -14.12 -28.23 3.885 -1.1E-04 -0.357 -0.713 5 1.49E-05 0.155 0.31

2.958 7E-05 -17.07 -34.1 5.828 -3.2E-05 3.501 6.98 7.5 -3.05E-04 0.228 0.572

3.944 -0.0194 -95.36 -191.2 7.770 -0.0047 -23.03 -46.11 10 -0.0049 7.421 14.78 -250 -200 -150 -100 -50 0 50 0 1 2 3 4 T o tal h ea t flu x Q z (W /m ^2 ) ΔT

Total heat flux

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18

Moreover, by observing Figure 15 and Figure 17, the moisture and the vapour fluxes have higher values close to the top surface as the evaporation rate is higher in these areas. Moreover, as is expected, the moisture is moving toward the top surface and the vapour downward to the bottom surface which is also illustrated in the vector plots of the moisture and the vapour fluxes (Figure 16 and Figure 18). Finally for the case

1a the gas velocity vector changes linearly (Figure 19).

Figure 15: Total moisture flux as a function of the temperature gradient applied in the z direction

Figure 16: Vector plot of the total moisture flux for y=5mm

Figure 17: Total vapour flux as a function of the temperature gradient applied in the z direction

Figure 18: Vector plot of the total vapour flux for y=5mm

Figure 19: Velocity vector as a function of the temperature gradient applied in the z direction

It is also interesting to observe the behaviour of the temperature in a surface. In case 1a where the gradient is applied only in the z direction the temperature in the surrounding four surfaces (x2, x1, y2 and y1) remain constant at their initial values. -2E-08 -2E-08 -1E-08 -5E-09 0E+00 5E-09 0 1 2 3 4 T o tal m o istu re f lu x (1 /(s* m ^2 )) ΔT

Total moisture flux

z=0mm z=0,986mm z=1,972mm z=2,958mm z=3,944mm -4E-06 -2E-06 0E+00 2E-06 4E-06 6E-06 8E-06 1E-05 0 1 2 3 4 T o tal v ap o u r flu x (k g /(s* m ^2 )) ΔT

Total vapour flux

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19

So the temperature values on the surfaces z2 and z1 will be influenced by the

temperature of the four surrounding surfaces (x2, x1, y2 and y1) since they have

common edges. As a result, when the system reaches steady state the temperature will not be homogeneous on the surface, with different values on the perimeter of the z2

and z1 surfaces (Figure 20). Moreover, the same behaviour appears in the total heat

flux (Figure 21). Finally similar phenomena exist in all the cases for the different variables and fluxes. This observation marks a clear inconsistency of the numerical solution with the defined boundary conditions. Overcoming or minimizing this effect would require a substantial modification of the computational grid with, however, a heavily increase of the computational load.

Figure 20: Temperature behaviour in surface z=3.944mm

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20

8.1.2. Case 1b: Temperature gradient in x the direction

In this case, temperature gradient is applied only in x the direction, where T2 is the

temperature on the right surface and T1 on the left surface of the cell and T2> T1.

Figure 22: Total heat flux as a function of the temperature gradient applied in the x direction

Figure 23: Total moisture flux as a function of the temperature gradient in x direction

Figure 24: Total vapour flux as a function of the temperature gradient applied in the x direction

Figure 25: Velocity vector as a function of the temperature gradient applied in the x direction

-160 -120 -80 -40 0 40 0 1 2 3 4 T o tal h ea t flu x Q x (W /m ^2 ) ΔT

Total heat flux

x=0mm x=1.943mm x=3.885mm x=5.828mm x=7.77mm -8E-09 -6E-09 -4E-09 -2E-09 0E+00 2E-09 0 1 2 3 4 T o tal m o istu re f lu x ( 1/( s* m^ 2 ) ΔT

Total moisture flux

x=0mm x=1.943mm x=3.885mm x=5.828mm x=7.77mm -5E-07 0E+00 5E-07 1E-06 2E-06 0 1 2 3 4 T o tal v ap o u r flu x (k g /(s* m ^2 )) ΔT

Total vapour flux

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21

8.1.3. Case 1c: Temperature gradient in y the direction

In case 1c temperature gradient is applied in y the axis, where T2 is the temperature on

the back surface and T1 on the front surface and T2> T1.

Figure 26: Total heat flux as a function of the temperature gradient applied in the y direction

Figure 27: Total moisture flux as a function of the temperature gradient applied in the y direction

Figure 28: Total vapour flux as a function of the temperature gradient applied in the y direction

Figure 29: Velocity vector as a function of the temperature gradient applied in the y direction

In the cases 1b and 1c, the same pattern is observed as in case 1a (temperature gradient in z direction). The fluxes show linear behaviour as the temperature gradient increases and have higher values close to the top and the bottom surfaces. Finally the velocity vectors have linear behaviour too (Figure 19, Figure 25 and Figure 29), but their slopes vary for the different cut planes, in cases 1a, 1b and 1c. This can be explained from the different geometry which has the unit cell in its three axes, and how the flutes influence the velocity of the gas.

-160 -120 -80 -40 0 40 0 1 2 3 4 T o tal h ea t flu x Q y (W /m ^2 ) ΔT

Total heat flux

y=0mm y=2.5mm y=5mm y=7.5mm y=10mm -8E-09 -6E-09 -4E-09 -2E-09 0E+00 2E-09 0 1 2 3 4 T o tal m o istu re f lu x (1 /(s* m ^2 )) ΔT

Total moisture flux

y=0mm y=2.5mm y=5mm y=7.5mm y=10mm -5E-07 0E+00 5E-07 1E-06 2E-06 2E-06 0 1 2 3 4 T o tal v ap o u r flu x (k g /(s* m ^2 )) ΔT

Total vapour flux

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22

8.1.4. Case 2a: Moisture gradient in z the direction

The moisture gradient is applied only in z the direction, where u2 is the moisture on

the top surface and u1 on the bottom surface and u2> u1.

Figure 30: Total heat flux as a function of the moisture gradient applied in the z direction

Figure 31: Total moisture flux as a function of the moisture gradient applied in the z direction

Figure 32: Total vapour flux as a function of the moisture gradient applied in the z direction

Figure 33: Velocity vector as a function of the moisture gradient applied in the z direction

-10 -8 -6 -4 -2 0 2 0 0,002 0,004 0,006 T o tal h ea t flu x Qz (W /m ^2 ) Δu

Total heat flux

z=0mm z=0,986mm z=1,972mm z=2,958mm z=3,944mm -8E-09 -6E-09 -4E-09 -2E-09 0E+00 2E-09 0 0,002 0,004 0,006 T o ta l m o istu re f lu x (1 /(s* m ^ 2 )) Δu

Total moisture flux

z=0mm z=0,986mm z=1,972mm z=2,958mm z=3,944mm -2E-06 -1E-06 0E+00 1E-06 2E-06 3E-06 4E-06 -4,34E-18 0,002 0,004 0,006 T o tal m o istu re f lu x (k g /(s* m ^2 )) Δu

Total vapour flux

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23

8.1.5. Case 2b: Moisture gradient in x the direction

The moisture gradient is applied only in x the direction, where u2 is the moisture on

the right surface and u1 on the left surface and u2> u1.

Figure 34: Total heat flux as a function of the moisture gradient applied in the x direction

Figure 35: Total moisture flux as a function of the moisture gradient in the x direction

Figure 36: Total vapour flux as a function of the moisture gradient applied in the z direction

Figure 37: Velocity vector as a function of the moisture gradient applied in the x direction

-1 -0,8 -0,6 -0,4 -0,2 0 0,2 0 0,002 0,004 0,006 T o ta l h ea t fl u x Q x (W /m^ 2 ) Δu

Total heat flux

x=0mm x=1,943mm x=3,885mm x=5,828mm x=7,77mm -8E-09 -6E-09 -4E-09 -2E-09 0E+00 2E-09 0 0,002 0,004 0,006 T o tal m o istu re f lu x (1 /(s* m ^2 )) Δu

Total moisture flux

x=0mm x=1,943mm x=3,885mm x=5,828mm x=7,77mm -2E-07 0E+00 2E-07 4E-07 6E-07 0 0,002 0,004 0,006 T o tal v ap o r f lu x (k g /(s* m ^2 )) Δu

Total vapor flux

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24

8.1.6. Case 2c: Moisture gradient in y the direction

The moisture gradient is applied only in y the direction, where u2 is the moisture on

the back surface and u1 on the front surface and u2> u1.

Figure 38: Total heat flux as a function of the moisture gradient applied in the y direction

Figure 39: Total moisture flux as a function of the moisture gradient applied in the y direction

Figure 40: Total vapour flux as a function of the moisture gradient applied in the y direction

Figure 41: Velocity vector as a function of the moisture gradient applied in the y direction

From the cases which are presented above (case 2a 2b and 2c), it is observed that the same pattern is followed when the moisture gradient is applied in different direction of the cell. More specific, the heat flux has lower values inside the unit cell, also has linearity behaviour and the flux decreases slightly as the moisture gradient increases. The velocity vectors in the x and the y direction slightly change values (10-7) as the moisture gradient is applied in the unit cell.

The moisture and the vapour move in different directions while the mass balance of the water remains constant. It is interesting to present how the moisture and the vapour move inside the cell (Figure 42 and Figure 43), where the moisture gradient applies in the y direction.

-1,2 -1 -0,8 -0,6 -0,4 -0,2 0 0,2 0 0,002 0,004 0,006 T o tal h ea t flu x Qy (W /m ^2 ) Δu

Total heat flux

y=0mm y=2,5mm y=5mm y=7,5mm y=10mm -7,E-09 -6,E-09 -5,E-09 -4,E-09 -3,E-09 -2,E-09 -1,E-09 0,E+00 1,E-09 0 0,002 0,004 0,006 T o tal m o istu re f lu x (1 /(s* m ^2 )) Δu

Total moisture flux

y=0mm y=2,5mm y=5mm y=7,5mm y=10mm -2E-07 0E+00 2E-07 4E-07 6E-07 8E-07 0 0,002 0,004 0,006 T o ta l v ap o r f lu x (k g /(s* m ^ 2 )) Δu

Total vapor flux

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25

Figure 42: Moisture flux for the moisture gradient in the y direction equal to 0.006, for x=3.885mm In Figure 42, it is showed only the moisture flux in y direction. Furthermore, the moisture moves only through the paper and the higher flux is close to the back surface of the cell where the high moisture value is applied.

Figure 43: Vapour flux for the moisture gradient in y direction equal to 0.006, for x=3.885mm

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26

8.1.7. Case 3a: Vapour concentration gradient in z the direction

The vapour concentration gradient is applied only in the z direction, where Cvp2 is the

vapour concentration on the top surface and Cvp1 on the bottom surface and Cvp2>

Cvp1.

Figure 44: Total heat flux as a function of the vapour concentration gradient applied in the z direction

Figure 45: Total moisture flux as a function of the vapour concentration gradient applied in the z direction

Figure 46: Total vapour flux as a function of the vapour concentration gradient applied in the z direction

Figure 47: Total velocity vector as a function of the vapour concentration gradient applied in the z direcdtion

-1 0 1 2 3 4 5 6 7 0 0,0005 0,001 T o tal h ea t flu x Q z (W /m ^2 ) ΔCvp

Total heat flux

z=0mm z=0,986mm z=1,972mm z=2,958mm z=3,944mm -1E-09 0E+00 1E-09 2E-09 3E-09 4E-09 0 0,0005 0,001 T o tal m o istu re f lu x (1 /(s* m ^2 )) ΔCvp

Total moisture flux

z=0mm z=0,986mm z=1,972mm z=2,958mm z=3,944mm -1E-05 -1E-05 -8E-06 -6E-06 -4E-06 -2E-06 0E+00 2E-06 0 0,0005 0,001 T o tal m o istu re f lu x ( kg /( s* m ^2 )) ΔCvp

Total vapour flux

(37)

27

8.1.8. Case 3b: Vapour concentration gradient in x the direction

The vapour concentration gradient is applied only in x the direction, where Cvp2 is the

vapour concentration on the right surface and Cvp1 on the left surface and Cvp2> Cvp1.

Figure 48: Total heat flux as a function of the vapour concentration gradient applied in the x direction

Figure 49: Total moisture flux as a function of the vapour concentration gradient applied in the x direction

Figure 50: Total vapour flux as a function of the vapour concentration gradient applied in the x direction

Figure 51: Total velocity vector as a function of the vapour concentration gradient applied in the x direction

-0,2 0 0,2 0,4 0,6 0,8 1 0 0,0005 0,001 T o ta l h ea t flu x Q x (W /m ^ 2 ) ΔCvp

Total heat flux

x=0mm x=1,943mm x=3,885mm x=5,828mm x=7,77mm -5E-10 0E+00 5E-10 1E-09 2E-09 2E-09 3E-09 0 0,0005 0,001 T o tal m o istu re f lu x (1 /(s* m ^2 )) ΔCvp

Total moisture flux

x=0mm x=1,943mm x=3,885mm x=5,828mm x=7,77mm -1,2E-05 -1,0E-05 -8,0E-06 -6,0E-06 -4,0E-06 -2,0E-06 0,0E+00 2,0E-06 0 0,0005 0,001 T o tal v ap o r f lu x (k g /(s* m ^2 )) ΔCvp

Total vapor flux

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28

8.1.9. Case 3c: Vapour concentration gradient in y the direction

The Vapour concentration gradient is applied only in y the direction, where Cvp2 is the

vapour concentration on the back surface and Cvp1 on the front surface and Cvp 2> Cvp1.

Figure 52: : Total heat flux as a function of the vapour concentration gradient applied in the y direction

Figure 53: Total moisture flux as a function of the vapour concentration gradient applied in the y direction

Figure 54: Total vapour flux as a function of the vapour concentration gradient applied in the y direction

Figure 55: Total velocity vector as a function of the vapour concentration gradient applied in the y direction

The cases 3a, 3b and 3c have the same pattern, where the total heat and the total moisture fluxes are influenced slightly when the vapour concentration gradient increases. Moreover, the influence of the velocity vector is even smaller.

Finally as it is expected from the theory and the equations of the model, the increase of the vapour concentration gradient increases the vapour flux. The effects are more obvious on the liners as the evaporation rate is higher. Figure 56 and Figure 57 are showing the vapour flux in two different applied gradients in z direction, 0.0006 kg/m3 and 0.001 kg/m3 and the vapour fluxes are greater when the vapour concentration gradient has higher values.

-0,2 0 0,2 0,4 0,6 0,8 1 1,2 0 0,0005 0,001 T o tal h ea t flu x Q y (W /m ^2 ) ΔCvp

Total heat flux

y=0mm y=2.5mm y=5mm y=7.5mm y=10mm -5E-10 0E+00 5E-10 1E-09 2E-09 2E-09 3E-09 0 0,0005 0,001 T o tal m o istu re flu x (1 /(s* m ^2 ) ΔCvp

Total moisture flux

y=0mm y=2.5mm y=5mm y=7.5mm y=10mm -1,2E-05 -1,0E-05 -8,0E-06 -6,0E-06 -4,0E-06 -2,0E-06 0,0E+00 2,0E-06 0 0,0005 0,001 T o tal v ap o u r flu x (k g /(s* m ^2 )) ΔCvp

Total vapour flux

(39)

29

Figure 56: Vapour flux, for vapour concentration gradient 0.0006 kg/m3 in z the direction

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30

8.1.10. Case 4a: Pressure gradient in z the direction

The pressure gradient is applied only in z the direction, where p2 is the pressure on the

top surface and p1 on the bottom surface and p2> p1.

Figure 58: Total heat flux as a function of the pressure gradient applied in the z direction

Figure 59: : Total moisture flux as a function of the pressure gradient applied in the z direction

Figure 60: Total vapour flux as a function of the pressure gradient applied in the z direction

Figure 61: Total velocity vector as a function of the pressure gradient applied in the z direction

-0,03 -0,02 -0,01 0 0,01 0,02 0,03 0 0,05 0,1 0,15 0,2 T o tal h ea t flu x Q z (W /( s* m ^2 )) Δp

Total heat flux

z=0mm z=0.986mm z=1.972mm z=2.958mm z=3.944mm -2E-11 -1E-11 -5E-12 0E+00 5E-12 1E-11 2E-11 0 0,05 0,1 0,15 0,2 T o tal m o istu re f lu x (1 /( s* m ^2 )) Δp

Total moisture flux

z=0mm z=0.986mm z=1.972mm z=2.958mm z=3.944mm -2E-08 -1E-08 -5E-09 0E+00 5E-09 1E-08 0 0,05 0,1 0,15 0,2 T o tal v ap o u r flu x ( kg /( s* m^ 2) ) Δp

Total vapour flux

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31

8.1.11. Case 4b: Pressure gradient in x the direction

The pressure gradient is applied only in x the direction, where p2 is the pressure on the

right surface and p1 on the left surface and p2> p1.

Figure 62: : Total heat flux as a function of the pressure gradient applied in the x direction

Figure 63: : Total moisture flux as a function of the pressure gradient applied in the x direction

Figure 64: Total vapour flux as a function of the pressure gradient applied in the x direction

Figure 65: Total velocity vector as a function of the pressure gradient applied in the x direction

-1E+04 0E+00 1E+04 2E+04 3E+04 4E+04 0 0,05 0,1 0,15 0,2 T o tal h ea t flu x Qx (W /s* m ^2 ) Δp

Total heat flux

x=0mm x=1.943mm x=3.885mm x=5.828mm x=7.77mm -2E-11 -1E-11 -5E-12 0E+00 5E-12 1E-11 2E-11 0 0,05 0,1 0,15 0,2 T o tal m o istu re f lu x (1 /( s* m ^2 )) Δp

Total moisture flux

x=0mm x=1.943mm x=3.885mm x=5.828mm x=7.77mm -5E-09 -3E-09 -1E-09 1E-09 3E-09 0 0,05 0,1 0,15 0,2 T o tal v ap o u r flu x ( kg /( s* m ^2 )) Δp

Total vapour flux

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32

8.1.12. Case 4c: Pressure gradient in y the direction

The pressure gradient is applied only in y the direction, where p2 is the pressure on the

back surface and p1 on the front surface and p2> p1.

Figure 66: Total heat flux as a function of the pressure gradient applied in the y direction

Figure 67: Total moisture flux as a function of the pressure gradient applied in the y direction

Figure 68: Total vapour flux as a function of the pressure gradient applied in the y direction

Figure 69: : Total velocity vector as a function of the pressure gradient applied in the y direction

Finally for cases 4a, 4b and 4c, the values of the moisture flux are kept constant (Figure 59, Figure 63 and Figure 67). The vapour fluxes have similar behaviour, as the values change insignificantly (Figure 60, Figure 64 and Figure 68).

Regarding the heat fluxes, when the gradient applies in z the direction the flux influences only the interior part of the cell and not the surfaces (Figure 58). On the contrary, in cases 4b and 4c the heat fluxes change as the gradient pressure increases more extensively on the surfaces of the unit cell (Figure 62 and Figure 66). The same behaviour appears on the velocity vector for these three cases (Figure 61, Figure 65 and Figure 69). It is observed that the values of the velocity vectors and the total heat flux of the case 4b and 4c are higher comparing to previous cases as the fluting influences the results.

0E+00 1E+04 2E+04 3E+04 4E+04 5E+04 6E+04 0 0,05 0,1 0,15 0,2 T o tal h ea t flu x Q y (W /s* m ^2 ) Δp

Total heat flux

y=0mm y=2.5mm y=5mm y=7.5mm y=10mm -2E-11 -1E-11 -5E-12 0E+00 5E-12 1E-11 2E-11 0 0,05 0,1 0,15 0,2 T o tal m o istu re f lu x (1 /( s* m ^2 )) Δp

Total moisture flux

y=0mm y=2.5mm y=5mm y=7.5mm y=10mm -6E-09 -4E-09 -2E-09 0E+00 2E-09 4E-09 6E-09 0 0,05 0,1 0,15 0,2 T o tal v ap o u r flu x ( kg /( s* m ^2 )) Δp

Total vapour flux

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33 8.1.13. Flow balance and accumulation

It is further interesting to investigate the balance of the cell which can be described as: ∑ ( ), where the deficit of the quantity X.

For the case 1 and gradient ΔT=4, the balance of the unit cell is going to be investigated. Firstly, it has to be mentioned that the signs of the calculated flows shows the direction of the flow which have been defined in the COMSOL Multiphysics® model.

Table 7: Accumulation factors for temperature gradient of ΔT=4 in z direction

Total heat flow (W)

-0.0138 -0.0159 0.00188 -0.00181 -4.458 e-4 4.530 e-4 -2.962 e-3

Total moisture flow(1/s)

-9.881e-13 -1.122e-12 -2.799e-14 2.743e-14 8.195e-15 -5.724e-15 -9.251 e-14

Total vapour flow (kg/s)

5.835e-10 6.742e-10 2.611e-12 -8.442e-13 -1.146e-11 1.123e-11 -1.130e-10

Momentum flow ((kgm)/s2)

-2.947e-22 2.805e-22 -1.229e-20 1.071e-20 -5.908e-21 4.756e-21 3.424e-22

From the accumulation factors of the temperature gradient (Table 7), it is observed that the numbers cannot be considered negligible and concluded that there is a difference in the energy balance of the total heat flow as change phase of the water occurs in the cell. Moreover this result is observed from the accumulation factor of the total moisture and total vapour flow which represent also the change phase of the water as it was already mentioned in 8.1.6.

8.1.14. Transport coefficients

The flux and the applied gradient are related through the transfer coefficient, which can be calculated from the slopes of the figures of the cases 1, 2, 3 and 4.

The transport coefficient is equal with the transport tensors kX,m (where X: the flux of

the variables T, u, Cvp, p and m : the direction which applies the gradient). More

specific for the heat flux when temperature gradient is applied:

(

),

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34

where Δ refers to the difference of the respective quantity over the unit cell.

With the same method the transport coefficients can be calculated for the other three variables for the heat fluxes which can be combined as:

From calculating the slopes of the heat fluxes as a function of the temperature gradient the transport coefficient is equal to:

(

)

In the same way, the transport tensors for the heat flux with gradient in moisture, vapour concentration and pressure are presented below:

( ) ( ) ( )

By observing the individual coefficients of the transport tensor, it is concluded that the higher values of this factor appears in the direction flux where the gradient is applied. These transport coefficients belong to the diagonal of the tensors and are named as

. Moreover, this conclusion is verified from values of

Table 3 which shows that the heat fluxes of the Qx and Qy are smaller comparing to

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35 8.2. Detailed model 2

As was mentioned in section 5.1.1, in this model fluxes apply only in the z direction in which temperature gradient is applied, while the other four surfaces of the unit cell have fixed values, equal to the initial and no fluxes.

8.2.1. Case 1a: Temperature gradient in z the direction

In this case the temperature gradient is applied only in z the direction, where T2 is the

temperature on the top surface and T1 on the bottom surface and T2> T1 and there is

flux only in the z direction.

Figure 70: Total heat flux as a function of the temperature gradient applied in the z direction

Figure 71: Total moisture flux as a function of the temperature gradient applied in the z direction

Figure 72 :Total vapour flux as a function of the temperature gradient applied in the z direction

Figure 73: Velocity vector as a function of the temperature gradient applied in the z direction

First of all, in this model the fluxes, in z direction, has linear behaviour as the temperature gradient increases. By comparing Figure 13, Figure 15, Figure 17 and Figure 19 of the detailed model 1 with Figure 70-Figure 73 of the detailed model 2, the fluxes of the model 1 vary in higher values. This is explained from the boundary conditions which are selected where there is insulation in the surfaces of the x and y axes. This conclusion is illustrated also on the Figure 74 and Figure 75.

-90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 0 1 2 3 4 T o tal h ea t flu x Q z (W /m ^2 ) ΔT

Total heat flux

z=0mm z=0,986mm z=1,972mm z=2,958mm z=3,944mm -2E-08 -2E-08 -1E-08 -5E-09 0E+00 5E-09 0 1 2 3 4 T o tal m o istu re f lu x (1 /(s* m ^2 )) ΔT

Total moisture flux

z=0mm z=0,986mm z=1,972mm z=2,958mm z=3,944mm -4E-06 -2E-06 0E+00 2E-06 4E-06 6E-06 8E-06 1E-05 0 1 2 3 4 T o tal v ap o u r flu x (k g /(s* m ^2 )) ΔT

Total vapour flux

(46)

36

Moreover, even if the side walls are insulated, the fluxes through the different cut planes vary, since the evaporation rate is higher in the two liners of the unit cell (Figure 70).

Figure 74: Detailed model 1: vector plot of the total heat flux for y=5mm.

Figure 75: Detailed model 2: Vector plot of the total heat flux for y=5mm.

In these two different models, the scale factor of the arrows is the same so the results can be compared. A significant amount of the heat comes in and out of the cell as it is observed in the Figure 74, which does not exist in the second model (Figure 75) as the values of the Qx and Qy are zero.

Moreover, in the detailed model 2, there is consistent boundary conditions and as is observed in Figure 76 the temperature of the top surface is the homogeneous.

Figure 76: Temperature behaviour in surface z=3.944mm

References

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