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Tampereen teknillinen yliopisto. Julkaisu 657 Tampere University of Technology. Publication 657 Timo Karvinen Pulsation Analysis of Paper Making Processes Thesis for the degree of Doctor of Technology to be presented with due permission for public examination and criticism in Konetalo Building, Auditorium K1702, at Tampere University of Technology, on the 12th of April 2007, at 12 noon. Tampereen teknillinen yliopisto - Tampere University of Technology Tampere 2007

ISBN 978-952-15-1741-9 (printed) ISBN 978-952-15-1769-3 (PDF) ISSN 1459-2045

i

ABSTRACT

Pulsation phenomena in two processes of a papermaking machine were studied. The processes are that taking place in the approach flow system and the headbox and the coating process. Pressure and velocity pulsation is created by pumps and screens in the approach flow system in the frequency range 5-100 Hz, which creates harmonic basis weight variation in paper in the machine-direction. Basis weight variation is a quality defect in paper and this variation in paper properties also affects the runnability of the paper machine. In the coating process a screw pump creates pulsation which can be seen as harmonic basis weight variation in coating in the machine-direction. Pulsation was modeled in the approach flow system and the headbox using the equations of fluid transients, which are the same as the equations of acoustics when friction can be neglected, which is the case in the approach piping. One- and three-dimensional solutions were compared assuming the structure around the fluid perfectly rigid. In the three-dimensional solution, which was obtained using a commercial FEM code, the fluid-structure interaction (FSI) was taken into account. In the coating process, in which it is extremely important to include friction, pulsation was modeled using the equations of fluid transients, the equations of pulsatile flow, and using a CFD code that solves the Navier-Stokes equations. Tapered slice channels of the headbox were analyzed in detail. It was found that the shape of the channel can affect the velocity oscillation amplitude and thus machine-direction basis weight variation at the channel outlet significantly. Changing the traditional linear shape to parabolic could decrease basis weight variation almost by one third for the same excitation amplitude. The fluid-structure interaction must be taken into account to correctly model pulsation in the approach piping and the headbox. The rigid and FSI solutions can yield large differences in the results. Laboratory measurements using particle image velocimetry confirm that FSI needs to be included in the analysis. Modeling of the approach piping and the headbox reveals that with FSI included in the analysis, many experimental findings of basis weight variation of paper can be explained. The relationship between pressure pulsation in the piping system and the machine-direction basis weight variation in paper was shown computationally. No simple relationship was found.

ii

In the paper coating process, modeling friction is extremely important. In the case of Newtonian fluid the approaches of harmonic fluid transients, the method of characteristics, and the analytical solution of pulsatile flow yield similar results because of the long time scale of the problem. In the case of non-Newtonian fluid, which is encountered in the paper coating, the solution is obtained by solving the Navier-Stokes equations with a CFD code. Also, the approach of harmonic fluid transients gives similar results for velocity pulsation. The relationship between pressure and velocity pulsation and basis weight variation was shown.

iii

PREFACE

This project started in August 2002. The present thesis would not have been produced without the actions of Mr. Hannu Lepomäki from Metso Paper, Inc., to whom I am very indebted. I asked him about the possibility to do a master’s thesis related to fluid dynamics and acoustics in the paper machine. A suitable subject was found. The aim was to find a relationship between pulsation in the approach piping and machine-direction basis weight variation in paper. The task was not thoroughly solved in the master’s thesis; there still remained problematic issues requiring further study. After receiving my master’s degree, I continued with the same theme in my doctoral work. This work has been done at the Institute of Energy and Process Engineering at Tampere University of Technology. I am grateful to my supervisor Docent Hannu Ahlstedt and to all of the Institute staff for providing guidance and excellent working conditions to accomplish this project. Especially Dr. Hannu Eloranta and Mr. Tero Pärssinen are commended for contributing their time and expertise when I was conducting the PIV laboratory measurements presented in Chapter 4. Metso Paper, Inc. is acknowleged for providing financial support. I am also grateful to many persons from the Rautpohja and Valkeakoski units of Metso, who are not mentioned here by name lest any be unintentionally left out, for providing support and data necessary for my research. The Graduate School Concurrent Mechanical Engineering and its director Prof. Erno Keskinen are acknowledged for the financial support and for the good education program. Especially the scientific discussions with Prof. Michel Cotsaftis from ECE Paris helped to complete the thesis. I am indebted to Prof. Jari Hämäläinen from the University of Kuopio and Prof. Timo Siikonen from Helsinki University of Technology for the examination of the manuscript. Their valuable comments led to several improvements. J.D. Donoghue is acknowledged for revising the language of the thesis. Finally, I would like to thank my father Prof. Reijo Karvinen, for the guidance he has provided throughout my studies to reach the highest academic degree. Tampere, April 2007 Timo Karvinen

iv

v

TABLE OF CONTENTS

ABSTRACT i

PREFACE iii

TABLE OF CONTENTS v

NOMENCLATURE vii

1 INTRODUCTION 1 1.1 Pulsation in approach flow system and headbox 1 1.2 Pulsation in paper coating 7 1.3 Earlier research on basis weight variation 7 1.4 Assumptions and prediction methods 10 1.5 Hydraulic machines 11 1.6 Objectives and organization of thesis 11

2 GOVERNING EQUATIONS 13 2.1 Equations of fluid transients 13

2.1.1 Equation of motion 13 2.1.2 Equation of continuity 15 2.1.3 Wall shear stress 18 2.1.4 Time scales 21 2.1.5 Solution of equations of fluid transients in time domain 22 2.1.6 Equations for oscillatory fluid transients 24

2.2 Pressure pulsation in one-dimensional tapered channel 25 2.3 Equations governing pulsation in three-dimensional case 25 2.4 Fluid-structure interaction 26

2.4.1 Equations of fluid-structure interaction in one dimension 28 2.5 Equations for pulsating laminar flow in slit 29

2.5.1 Newtonian flow 29 2.5.2 Non-Newtonian flow 30

2.6 Solution of equations governing pulsation 32

3 PULSATION IN TAPERED CHANNELS 34 3.1 Rigid, one-dimensional model of pipeline 34

3.1.1 Pipe/channel sections 34 3.1.2 Cross-section changes in piping and headbox 35

vi

3.1.3 Assembly of different components 35 3.2 Analysis of tapered slice channel 36 3.3 Analytical solutions of Webster’s equation 37

3.3.1 Linearly tapered channel 37 3.3.2 Exponentially tapered channel 38 3.3.3 Other channel shapes 39

3.4 Comparison of different channel shapes 39 3.5 Comparison of analytical and numerical solutions 42 3.6 Effect of channel dimensions on volume flow rate oscillation in linear channel 42 3.7 Effect of air on pulsation 43 3.8 Effect of vanes on pulsation 45

4 PULSATION IN PAPER MACHINE APPROACH PIPING AND HEADBOX 47 4.1 Paper machine measurements 47 4.2 Volume flow rate oscillation in long rigid pipeline 48 4.3 Rigid vs. FSI solution in straight pipe 48 4.4 1D rigid model equations for paper machine system 50 4.5 Laboratory measurements and analysis of tapered channel – manifold system 52 4.6 Real paper machine model, headbox width 1 m 56 4.7 Real paper machine model, headbox width 10 m 67 4.8 Computation requirements and mesh density 71 4.9 Comparison of modeling results and paper machine measurement data 73 4.10 System design aspects 74

5 PULSATION IN PAPER COATING PROCESS 76 5.1 Newtonian flow 77 5.2 Non-Newtonian flow 78

6 CONCLUSIONS 82

REFERENCES 86

APPENDIX A 91

vii

NOMENCLATURE Latin letters

A cross-sectional area, amplitude in natural mode

Ai constant of unsteady friction weighting function

a wave speed

Bi constant of unsteady friction weighting function

BW basis weight

b slope of linear channel

C unknown constant, capacitance

C damping matrix

C* parameter related to unsteady friction coefficient

D diameter

E Young’s modulus

e structural thickness

F transfer matrix

f frequency, Moody friction factor

f force vector

G Shear modulus

g gravitational acceleration

H pressure (piezometric) head

h constant of exponential channel, height

Is second moment of inertia

i imaginary unit, channel section

J0 Bessel function of first kind

Js polar second moment of area

K bulk modulus

K stiffness matrix

viii

k wave number, consistency iendex

k3 unsteady friction coefficient of Brunone’s model

L length, inertance

l length of channel section, natural mode index

M Mach number, bending moment

M mass matrix

m constant of linear channel, natural mode index, exponent of diameter in

power law

ma mass of air per cubic unit of volume

n exponent depending on flow type, power-law index, natural mode

index

p pressure

px pressure gradient amplitude

Q volume flow rate, shear force

R resistance factor, gas constant, radius

Re Reynolds number

T temperature, traction at fluid-structure interface

Td radial diffusion time scale

t time

t´ dummy variable

U fluid velocity, voltage

u pipe/structure displacement

u& pipe/structure velocity

u&& pipe/structure acceleration

V average fluid velocity

W weighting function, channel width

x x-coordinate

Y0 Bessel function of second kind

ix

y channel height, y-coordinate, displacement of structure

Z impedance

ZC characteristic impedance

z z-coordinate

Greek letters

Γ dimensionless parameter related to wall shear stress

γ propagation constant

γ& shear rate

Δp pressure amplitude

ΔV velocity amplitude

Δt time step (method of characteristics discretization)

Δx pipe/channel reach length (method of characteristics discretization)

η dynamic viscosity

κ2 shear coefficient of pipe wall material

ν kinematic viscosity, Poisson constant

θ pipe rotation angle

θ& rotation velocity of pipe

ρ density

σ normal stress, decaying factor

τ time scale

τw wall shear stress

ω angular frequency

ξ Darcy-Weisbach friction factor

ζ time scale parameter, inertia factor

x

Subscripts

1 upstream

2 downstream

a air

abs absolute

av average

c cross-section change

D downstream

f fluid

g gas

liq liquid

m manifold

n normal

out outlet

p pipe

s structure

t tapered channel

U upstream

w wall

ws quasi-steady wall shear stress

wu unsteady wall shear stress

Superscripts

* dimensionless quantity

xi

Acronyms

CD Cross-direction

CFD Computational fluid dynamics

FEM Finite element method

FSI Fluid-structure interaction

MD Machine-direction

MOC Method of characteristics

xii

1

1 INTRODUCTION In the paper machine, there are numerous different types of fluid flow phenomena at different parts of the machine. The paper machine and its unit operations are shown in Fig. 1.1. This study is concentrated on the pulsation phenomenon. Pressure and velocity or volume flow rate oscillations, which are disturbances in the flow, affect paper quality and machine runnability. Modeling of pulsation is important to understand how pulsation affects paper quality and how pulsation could be prevented. Two different processes involving the pulsation phenomenon are covered, the first one being that taking place in the approach flow system and the headbox and the second one the paper coating process.

1.1 Pulsation in approach flow system and headbox The approach flow system of the paper machine is an important part of the process of paper making. The stock, which is a mixture of water and fibers, is prepared in the approach flow system (Fig. 1.2) such that it is ready to be fed to the headbox. The main operations of the approach flow system are (Paulapuro, 2000):

- Dilution of stock to headbox consistency - Removal of product- and production-disturbing contaminants (solids

and air) - Conditioning with chemicals and additives - Feeding the headbox - Supply of additional water for paper machine cross-profile control in

case of headbox dilution system However, the approach flow system also creates disturbances that can be seen in the end product, paper. These disturbances include variation in flow speed, pressure, stock consistency, uneven distribution of chemicals and additives, and many others. Many of these disturbances affect the formation of paper. This study is concentrated on the pulsation phenomenon that causes short-term variation in the basis weight of paper in the machine-direction, shown in Fig 1.3, in the frequency range of approximately 5-100 Hz.

2

Fig. 1.1. Light weight coated paper machine. Published with permission of Metso Paper, Inc.

3

Fig. 1.2. Approach flow system of paper machine. 1. Machine chest, 2. wire pit, 3. stock fan pump, 4. cleaner banks, 5. deaeration tank, 6. vacuum pump, 7. headbox feed pump, 8. screen(s), 9. dosage points (Paulapuro, 2000). Published with permission of Fapet Oy. The most significant disturbance causing machine-direction basis weight variation in paper is harmonic pressure pulsation created by the headbox feed pump and the screen(s) (components 7 and 8 in Fig. 1.2). The disturbances are created at the rotational frequency of the hydraulic machine and their multiples. The generation mechanism of pulsation is not discussed in detail in the study. Information on pump- and screen-generated pressure pulsation can be found e.g. in articles of Guelich (1992) and Feng (2005). Pulsation may originate also from other components in the approach flow system or the dilution system of the headbox, which is shown in Figs. 1.4 and 1.5, but pulsation created by sources other than the headbox feed pump or the screens is in most cases quite insignificant. Other types of variations are basis weight variation in the cross-direction and residual variation. Pressure pulsation means that there is also velocity and volume flow rate variation in the flow. Pulsation travels from the approach flow system to the headbox with the speed of sound in the stock. When the fluid is injected from the headbox into the paper machine wire, the volume flow rate variation causes a non-uniform rate of fiber discharge from the headbox.

4

Fig. 1.3. Components of basis weight variation. Published with permission of Metso Paper, Inc. This variation can be seen in the final product as thickness variation in paper, i.e. basis weight (mass / unit area) variation. Paper usually has an end use either as a covering or packaging material or as a medium of information transfer by being the surface receiving a printed image. The strength properties are usually dependent on the basis weight, and thus the more uniform the basis weight the better the strength properties. Basis weight irregularities, whether on a large or small scale, should therefore be avoided using a combination of equipment design and operating skill. Additional to the quality effects are the effects of basis weight on the runnability of the machine. To minimize the number of breaks on the machine, dips in the web strength caused by basis weight and moisture variation must be avoided and thus uniformity of basis weight with time is required. As the speed of paper machines continues to increase, undisturbed flow conditions are becoming more and more important in the approach flow system. Usually the interest of research in the field of fluid transients is to avoid having a catastrophic failure in the piping system for example in a nuclear power plant or a water main. The aim of this study is to model pressure pulsation with a view to improving product quality. Of course, if resonance were to occur in a paper machine piping system, a breakdown could happen, but such an event is not very likely because the excitation amplitudes are relatively low. One of the most important components in the paper machine is the headbox, shown in Figs. 1.4 and 1.5. The main function of the headbox is to distribute the stock evenly across the wire section. The stock is spread onto the wire by accelerating the fluid in a tapered channel.

5

Fig. 1.4. Structure of headbox. 1. Manifold, 2. dilution system, 3. manifold tube bank, 4. equalizing chamber, 5. turbulence generator, 6. slice channel, 7. guide vanes, 8. slicebar. Published with permission of Metso Paper, Inc.

Fig. 1.5. Top view of headbox flow elements. Published with permission of Metso Paper, Inc. Fig. 1.6 is an example of the measured machine-direction basis weight spectrum of paper. Several peaks can be seen in the spectrum and many of them can be detected in the approach flow system by measuring pressure pulsation.

6

Fig. 1.6. Spectrum of variation in machine-direction basis weight.

Fig. 1.7. Principle of manifold flow spreader. The basis weight variation peaks shown in Fig. 1.6 are extremely high because pressure pulsation was created on purpose to study the relationship between pressure pulsation and basis weight variation. Variation of this magnitude would result in very poor quality of paper. Before the headbox, the flow is distributed evenly over the machine width from a circular pipe. This is achieved with a manifold shown in Fig. 1.7. The flow direction is changed 90 degrees in the manifold. In a real paper machine there are small tubes or channels after the manifold, shown in Figs 1.4 and 1.5. Their function is to create a pressure drop and to even out the mean velocity differences in the cross-direction. The turbulence just before the slice channel has the same function and it also increases the turbulence level, which improves the fiber orientation of paper. If the manifold has a linear taper, the mean velocity and volume flow rate on the front side (i.e. the edge further from the inlet flow edge) of the machine is typically 10 % higher than on the back side (Syrjälä, 1986). The shape of the manifold can be optimized to obtain a uniform mean velocity distribution in the cross-direction

7

(Hämäläinen, 2000). Also, the opening of the slice bar affects the cross-direction basis weight profile. The admissible variation in basis weight in the machine-direction is of the magnitude of 0.25 % of the mean basis weight, depending slightly on the paper grade.

1.2 Pulsation in paper coating In paper coating the coating material, which is a mineral-based, highly viscous fluid, is pumped with the aid of a screw pump. The principle of the paper coating process is shown in Fig. 1.8. The fluid flows from the piping into a manifold from where it is spread evenly to a very narrow slit having the width of the machine wire. After the slit, the coating paste flows on a slide onto the paper. The screw pump creates pulsation, which can be seen as the volume flow rate oscillation in coating paste and finally as the periodic coating thickness oscillation in paper. The problem is similar to that of the variation in base paper basis weight described above, but the fluid used is different. The flow is non-Newtonian and the behavior of the fluid differs from that of e.g. water or air. The flow behavior of the water-fiber mixture is also non-Newtonian, but if we are studying pressure wave propagation in the approach flow system, the non-Newtonian nature, which is related to friction, becomes irrelevant. An examination of the coating process is included in this study, not because of the non-Newtonian nature of the flow, but because of the same phenomenon as encountered in the approach flow system and the headbox. The problem can possibly be described with the same equations of fluid transients as for the pulsation problem in the approach flow system and the headbox.

1.3 Earlier research on basis weight variation The most significant research articles on the basis weight variation in paper were published from the end of the 50’s to the early 80’s. These articles include the work of Mardon (1958), Moen (1977), Parker (1976, 1977), and Hauptmann (1981), to mention only the most significant ones. Since then, very little new information has been published concerning pulsation-generated basis weight variation.

8

Fig 1.8. Principle of paper coating machine.

Fig. 1.9. Wet end barring on fourdrinier machine. Published with permission of Metso Paper, Inc. Pulsation can be a severe problem in fourdrinier machines, which are machines where the stock is spread from the headbox to a long horizontal wire, where the removal of water is initiated. The pulsation problem in fourdrinier machines has been explained by means of the wet end barring theory (Parker, 1977), according to which the velocity pulsation is amplified on the wire, which is shown in Fig 1.9. The velocity oscillates in the headbox jet and is spread onto the wire. On the wire, the faster moving stock will catch up with the slower moving stock, causing troughs and peaks to be formed, as shown in Fig. 1.10. At a certain instant after the stock leaves the headbox, the relative motion will cease and the wave profile will be found in paper.

9

Fig. 1.10. Development of waves on the wire (Parker, 1977). The frequency at which this phenomenon is greatest depends on the machine speed and it was shown that the maximum effect occurs at the frequency

ghl

Vf4

2= (1.1)

where f is the frequency, V the machine speed, l the distance between the headbox outlet (slice) and the point at which the stock sets, and h is the depth of stock on the wire. This frequency is typically below 20 Hz, a frequency range in which the basis weight variation usually is strongest. With the developments in paper machines the pulsation problem has become less significant. Earlier, the machines were fourdrinier-type machines. Nowadays, the fast-running machines, which can have a speed of 2000 m/min (30 m/s), have a twin-wire gap former right after the headbox instead of the long wire and the waves do not have time to develop, with the result that the above-mentioned problem does not exist. A higher pressure pulsation amplitude is allowed in the headbox. There are also pulsation dampers used with some headboxes. They have the maximum attenuation at the natural frequency of the damper, but other frequencies are less attenuated. The pulsation dampers are not included in the analysis in this study. Pressure pulsation analysis in the past was done using simple acoustics and fluid dynamics models. The development of numerical techniques and the increase in the speed of computers have made it possible to analyze these complicated systems more accurately.

10

It is well known that high basis weight variation is related to the excitation coinciding with some acoustic natural frequency of the piping system. Also the importance of mechanical vibration is recognized. If a high vibration amplitude is found at some part of the piping or in the headbox, basis weight variation is very probably found in paper at that frequency (Seifert, 1980). Acoustical and structural modes can also be coupled. The fluid-structure interaction has been mentioned as significant in the studies of Perrault (1984) and Lamoureux (1991). No pulsation analysis of the full approach piping and the headbox has been done, according to the author’s knowledge. The author has found no published information related to pulsation in the paper coating process and no pulsation analysis has been conducted according to the author’s knowledge.

1.4 Assumptions and prediction methods In this study the interest is on the low frequency excitation, the highest frequency of interest being approximately 100 Hz. In this range the pressure waves propagate in plane waves only and the mechanical vibrations in the pipes propagate in longitudinal, flexural and torsional waves only. The flow speed in the pipes and channels studied is much smaller than the critical velocity at which the pipes become unstable (Blevins, 1993). Also, the Mach number of the flow is small and the effect of the mean flow can be ignored (Wylie, 1993). The phenomena studied are steady harmonic. Time domain methods (e.g. the method of characteristics) and frequency domain methods (the impedance method, the finite element method (FEM)), in principle lead to the same results or at least to results that can be easily converted to each other by means of the Fourier transform. For the steady harmonic phenomena the frequency domain solution is more practical and computationally efficient. One-dimensional equations can be used when analyzing long pipelines and channels. They should be used whenever possible because of ease of use and the low computational cost. If three-dimensional elements are present in the system, the solution becomes more complicated and the three-dimensional equations must be employed. In the analysis, it is assumed that for gap former machines the basis weight variation is proportional to the velocity variation at the channel outlet such that

BWBW

VV Δ

=Δ (1.2)

11

where V is the mean velocity, BW the basis weight (g/m2), and the terms with the sign Δ stand for the amplitude of oscillation.

1.5 Hydraulic machines As mentioned above, hydraulic machinery such as pumps and screens create pulsation. The pulsation generation mechanisms are complex and are not discussed in this study. An interested reader may consult the paper of Guelich (1992) or Rzentkowski (2000). The hydraulic machinery does not only act as the source of pulsation. The machine attenuates pulsation coming from upstream of the approach flow system. To determine the effect of the hydraulic machine on pulsation, a precise computational model should be built to obtain the attenuation characteristics spectrum. Another possibility is to measure the acoustical characteristics of the machine. The pump can be represented as an acoustic two-port and a transfer matrix can be obtained that represents the transmission and generation of pressure waves between the pump inlet and outlet (de Jong, 1994). This is almost an impossible task to perform for pumps that are in use in an industrial process. Also, the system can have a great influence on pressure pulsation created by the pump (Guelich, 1992). In this study the pressure pulsation amplitude is assumed to be known right after the hydraulic machinery and the propagation of pressure waves through the hydraulic machines is not modeled and included in the analysis.

1.6 Objectives and organization of thesis As mentioned above, pulsation problems in two different processes of papermaking are covered in this study. Pulsation in the approach piping and the headbox is analyzed to obtain a relationship between pressure pulsation in the piping system and machine-direction basis weight variation in paper. The tapered slice channel of the headbox is analyzed in detail to see how the channel shape and dimensions affect velocity pulsation at the channel outlet. In the coating process a relationship between pressure pulsation and basis weight variation will be obtained. The overall goal of the study could be summarized as an attempt to improve the existing system and design to produce better quality paper. In Chapter 2 the theory and equations governing pulsation are presented. The equations of fluid transients are derived for a pipe or channel with non-constant cross-section and different aspects related to these equations are

12

discussed, especially the effect and modeling of friction and the time scale. Furthermore, the equations for pulsatile viscous flow are presented and the equations and the procedure for including the fluid-structure interaction in the analysis are shown. The tapered slice channel is studied in Chapter 3. Velocity oscillation in channels of different shape is compared and the effect of channel dimensions, the wave speed, and the guide vanes on pulsation is studied. Chapter 4 presents the analysis and results for the approach piping and the headbox. Laboratory model measurements and modeling results are compared. The analysis and results of the coating process are presented in Chapter 5. The results and conclusions are summarized in Chapter 6.

13

2 GOVERNING EQUATIONS The equations governing the pulsation phenomenon are presented in this chapter. The equations of motion and continuity of fluid transients are derived for a pipe or channel of non-constant cross-section. The solution of these equations is presented both in the frequency and time domains. The effect of friction and the time scale on the solution of these equations is discussed. The wave equation and Webster’s equation of the acoustics governing pulsation in channels of non-constant cross-section are derived from the equations of fluid transients. The equations of fluid-structure interaction are presented as well as the manner in which the fluid and the structure can be coupled. Finally, the equations governing viscous pulsatile flow in a narrow slit are presented both for Newtonian and non-Newtonian fluid.

2.1 Equations of fluid transients The equations of momentum and continuity are derived first for a pipe of non-constant cross-section using a procedure similar to that of Wylie (1993).

2.1.1 Equation of motion The free-body diagram for the equation of motion is shown in Fig. 2.1. The free body of fluid with the density ρ has a cross-sectional area A and a length of dx. The fluid element has an average cross-sectional velocity of V(x, t) and an average pressure p(x, t) at the centerline at the time t. The area is a function of x, which is the coordinate distance along the axis of the tube. The tube is inclined to the horizontal at the angle α. The forces on the free body in the x direction are the surface contact normal pressures on the transverse faces and shear and pressure components on the periphery. In addition, gravity has an x component. The shear stress τw acts in the negative x direction.

14

Fig. 2.1. Free-body diagram for application of equation of motion.

The sum of the forces on the fluid element shown in Fig. 2.1 is equal to its mass times its acceleration

( )

DtDVAdx

gAdxdxDdxxAdx

xppdxpA

xpApA w

ρ

αρπτ

=

−−∂∂

⎟⎠⎞

⎜⎝⎛

∂∂

++⎥⎦⎤

⎢⎣⎡

∂∂

+− sin21

(2.1) where D is the pipe diameter and

txV

DtD

∂∂

+∂∂

= (2.2)

is the total derivative with respect to time. By omitting the small quantity containing the term ( )2dx and simplifying, we obtain

0sin =+++∂∂

DtDVAgAD

xpA w ραρπτ (2.3)

The wall shear stress τw in Eq. (2.3) is the sum of the quasi-steady flow pipeline resistance τws and the pipeline inertance τwu accounting for liquid inertia, or

15

wuwsw τττ += (2.4) In the quasi-steady approach, the term τwu is assumed to be equal to zero. The pipeline shear stress is based on the Darcy-Weisbach equation and is

8

VVwsw

ξρττ == (2.5)

The quasi-steady friction factor ξ is calculated from the Hagen-Poiseuille law for laminar flow and from the Colebrook-White formula for turbulent flow. Using Eqs. (2.2) and (2.5), Eq. (2.3) takes the form

02

sin1=++

∂∂

+∂∂

+∂∂

DVV

gtV

xVV

xp ξ

αρ

(2.6)

which is valid for converging or diverging flow. If the Mach number

aVM /= of the flow is small, where a is the wave speed in fluid, the term xVV ∂∂ / is small compared to the other terms in Eq. (2.6). Furthermore, if we

have a zero slope in the pipe, Eq. (2.6) takes the form

02

1=+

∂∂

+∂∂

DVV

tV

xp ξ

ρ (2.7)

which is the well-known equation of motion of fluid transients.

2.1.2 Equation of continuity A control volume of length dx at time t is shown in Fig. 2.2. The law of the conservation of mass requires that the rate of mass inflow into the control volume is equal to the time rate of increase of mass within the control volume, or

( ) ( )t

AdxdxxAV

∂∂

=∂

∂−

ρρ (2.8)

16

Fig. 2.2. Control volume for continuity equation.

The expansion of Eq. (2.8) yields

0=∂∂

+∂∂

+∂∂

+∂∂

+∂∂ dx

tAdx

tA

xAVdx

xAVdx

xVA ρρρρρ (2.9)

and after simplifying

011=

∂∂

+∂∂

+∂∂

+∂∂

+∂∂

ttA

AxV

xA

AV

xV ρ

ρρ

ρ (2.10)

Eq. (2.10) can be put in the form

011=

∂∂

++xV

DtD

DtDA

Aρ

ρ (2.11)

with the use of total derivative in Eq. (2.2). Eq. (2.11) holds for converging or diverging pipes and it is also valid for rigid or highly deformable tubes as no simplifications have been made. The second term in Eq. (2.11) takes into account the compressibility of the fluid. The bulk modulus of elasticity K of the fluid is defined as (Wylie, 1993)

ρρ /Δ

Δ=

pK (2.12)

The wave speed of fluid is defined as (Wylie, 1993)

17

( )( )pAAKKa

ΔΔ+=

//1/ ρ (2.13)

For a thick-walled pipe the term pA ΔΔ / is very small and

ρKa = (2.14)

is the acoustic speed of a small disturbance in an infinite fluid. From Eq. (2.12) we obtain

K

DtDpDtD //=

ρρ (2.15)

This substitution excludes thermodynamic effects and limits its use to slightly compressible fluids only, i.e. the density change is small. By substituting Eq. (2.15) into Eq. (2.11) and using the equation of wave speed (2.14), we obtain

0112 =

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

+∂∂

+∂∂

xV

tp

xpV

atA

AxA

AV

ρ (2.16)

The cross-sectional area does not change with time in a rigid pipe and thus 0/ =∂∂ tA in Eq. (2.16). By combining the equation of motion and the continuity equation, it can be shown that in that combined equation there is a term ( )21/ Mxp −∂∂ (Wylie, 1993). This term is very small if the Mach number is small and the term containing xp ∂∂ / can omitted without loss of accuracy. With these simplifications, the equation of continuity (2.16) becomes

012 =

∂∂

+∂∂

+∂∂

xA

AV

tp

axV

ρ (2.17)

which is valid for converging or diverging pipes. If the pipe cross-section is constant, we may neglect the term with xA ∂∂ / and Eq. (2.17) becomes

012 =

∂∂

+∂∂

tp

axV

ρ (2.18)

18

The equation of motion (2.7) and the equation of continuity (2.18) are together the classical equations of fluid transients for a pipe of constant cross-section.

2.1.3 Wall shear stress It was indicated above that the wall shear stress can be expressed as the sum of quasi-steady and unsteady shear stresses as shown in Eq. (2.4). To determine the importance of the wall shear stress term in Eq. (2.3) (with zero slope), it can be cast in non-dimensional form

0/2

**

*

*

*=⎟

⎠⎞

⎜⎝⎛++

∂

∂+

∂

∂w

daL

TM

DL

tV

xp τζξζ (2.19)

where the superscript * denotes the dimensionless quantities with

aVpp ρ/* = the pressure, atx =* the x-coordinate, aVV /* = the velocity,

)//(* aLtt ζ= the time, )8//( 2* Vww ξρττ = the shear stress, and where ζ is a positive parameter related to the pressure transient time scale, L the pipe length, and ν/2DTd = (ν the kinematic viscosity) the radial diffusion time scale of the boundary layer across the pipe diameter, which is related to the transient induced radial velocity gradient. According to Eq. (2.19), the wall shear stress depends on the magnitude of the dimensionless parameter (Ghidaoui, 2005)

⎟⎠

⎞⎜⎝

⎛+=aL

TM

DLΓ d

/2ζξζ (2.20)

If Γ is significantly smaller than 1, the friction becomes negligible and τw can safely be set to zero. That is the case e.g. in a short pipeline. On the other hand, in highly viscous flows or in pipes or channels of small diameter, Γ is much greater than unity and the effect of friction is extremely important. The modeling of wall friction is essential for practical applications that demand transient simulation when the simulation time is long, i.e. ζ is large. The modeling of wall shear stress and unsteady friction models are the subject of extensive research in the water hammer research community, see for example the references (Vardy, 1996, 1997, 2003), (Brunone, 2000), and (Ghidaoui, 2002, 2005). There are two different approaches for modeling the unsteady wall shear stress τwu. One of them is the approach proposed by Zielke

19

(1968) in which the pipeline inertance is related to the local acceleration of flow by means of the weighting function W(t). In this approach the unsteady wall shear stress is based on the analytical solution of the unidirectional flow equations and is

( ) ( )∫ ′′−′∂∂

=t

wu tdttWttV

Rt

0

2)( νρτ (2.21)

where R is the pipe radius and t ′ a dummy variable. The weighting function is

( ) ( )∑=

−=5

1

/ 2

i

RtAietW ν for 02.02 ≥R

tν

( ) ∑=

−

⎟⎠⎞

⎜⎝⎛=

6

1

22

2i

i

iR

tBtW ν for 02.02 <R

tν (2.22)

{ }{ }0.351563-0.396696,0.9375,1.057855,1.25,-0.282095,

322.5544218.9216,135.0198,70.8493,26.3744,==

i

i

BA

New developments of the weighting function model also exist (Vardy, 1996, 2003). The other approach in unsteady friction modeling is the use of the simple model proposed by Brunone (1991). The unsteady shear stress is defined as

⎟⎠⎞

⎜⎝⎛

∂∂

−∂∂

=xVa

tVRkwu 23

ρτ (2.23)

where the correct value for k3 is obtained by comparing the calculations with the measurements and it is usually obtained by trial and error (Abreu, 2004). Vardy (1996) proposed an equation for calculating the coefficient from

2

*

3Ck = (2.24)

where *C is a coefficient depending on the instantaneous mean flow and is

00476.0* =C (2.25)

20

for laminar flow and

)/3.14log(

*05.0

41.7ReRe

C = (2.26)

for turbulent flow where the Reynolds number is Re = VD/ν. An exponential law may be used in the quasi-steady friction model. The term

DVV 2/ξ in the equation of motion (2.7) is replaced in a power-law by the term (Wylie, 1993)

m

n

D

VkV 1−

(2.27)

where k is the consistency index, n the power-law index and m the exponent of diameter. Let us consider a case in which friction can be neglected, i.e. Γ << 1. The equation of motion (2.7) takes the form

01=

∂∂

+∂∂

tV

xp

ρ (2.28)

Let us take the partial derivative of Eq. (2.28) with respect to x. Next we take the partial derivative of Eq. (2.18) with respect to t. Next, the latter equation is deducted from the former one, which results in

012

2

2

22

2

2=

∂

∂−

∂∂∂

−∂∂

∂+

∂

∂

tp

atxV

txV

xp ρρ (2.29)

The mixed derivatives disappear from Eq. (2.29) and we obtain the wave equation of acoustics (Morse, 1948)

012

2

22

2=

∂

∂−

∂

∂

tp

axp (2.30)

Thus, it has been shown that the one-dimensional wave equation and the equations of fluid transients are the same in the absence of friction for a pipe

21

of constant cross-section. As a matter of fact, fluid transients and acoustics involve exactly the same method and bear the same equations in some cases. In the literature, these two approaches that both study wave propagation, have become differentiated. It has been shown by the author of this study (Karvinen, 2006) that both approaches give identical results for pulsation in tapered channels. The effect of viscosity is important in viscous fluids such as oil, in high frequency applications, and in narrow pipes or channels. The pipes and tapered channels analyzed in this study using the theory of fluid transients belong to the category in which Γ << 1 (pipe length is of order of few meters) and thus the friction term can be omitted. However, the analysis of the rectangular slit employed in the study requires the use of friction, because Γ >> 1.

2.1.4 Time scales To determine which method to use for solving an unsteady flow problem, one needs to consider the time scales. If the transient event occurring in the system is much slower than the time scale of the solution method, it may be more convenient to resort to a different solution method. When the valve at the downstream end in a long pipeline with a reservoir upstream is opened slowly (e.g. in 1 minute), the problem can be solved using the rigid water column theory instead of employing the equations of fluid transients. The rigid water column theory assumes the fluid to be incompressible and the pipe walls completely rigid. This method includes only the equation of motion (2.7). If the valve is rapidly closed in a pipeline (e.g. in less than one second), the rigid water column theory is incapable of predicting the pressure rise accurately and the equations of fluid transients must be employed. The reason for this lies in the different time scales of the cases described above. The solution method to be used must have a time scale of the same order as the excitation in the system to be able to record all the characteristics of the phenomenon. If even smaller time scale phenomena need to be investigated than those which the equations of fluid transients are able to record, the fluid-structure interaction must be taken into account (Tijsseling, 2004). The typical time scale τ in turbulent flow for the rigid water column theory is

22

VD

ξτ 2= (2.31)

whereas the time scale of water hammer for the open-open ends pipe condition is

aL2

=τ (2.32)

For laminar flow the time scale of the rigid water column theory is obtained from the equation of continuity (2.7) by substitution of eR/64=ξ VD/64ν= and is

ν

τ32

2D= (2.33)

So far we have discussed the short time scale of unsteady flow which refers to the pressure wave travel time. There is also another kind of time scale in unsteady pipe flow. The longer time scale is the time of diffusion of the boundary layer from the pipe wall to the pipe core. The diffusion time scale Td is proportional to the kinematic viscosity and the pipe diameter. This time scale becomes important if the viscosity is high or the pipe diameter small. Pressure pulsation in the approach piping of the paper machine has the time scale of the same order, the reciprocal of frequency, as the fluid transient time scale, and that can be analyzed with the equations of fluid transients but not with the rigid water column theory. For the coating process, the rigid column theory would be sufficient. The pulsation cycle time is of the order of 0.01-0.1 s, whereas the rigid water column theory time scale for laminar flow of coating paste is of the order of 1⋅10-5 s, which is less than the diffusion time scale for the flow case.

2.1.5 Solution of equations of fluid transients in time domain The method of characteristics (MOC) is generally considered the most numerically efficient method for solving the equations of fluid transients (Wylie, 1993). Eqs. (2.7) and (2.18) are transformed into two pairs of ordinary differential equations, for positive and negative characteristics C+ and C-:

23

+

⎪⎪⎭

⎪⎪⎬

⎫

+=

=++C

adtdx

dtDVV

dpa

dV 02

1 ξρ

(2.34)

−

⎪⎪⎭

⎪⎪⎬

⎫

−=

=+−C

adtdx

dtDVV

dpa

dV 02

1 ξρ (2.35)

Eqs. (2.34) and (2.35) are integrated along the characteristics lines. The trapezoidal rule is used in the evaluation of the friction term, a procedure which is of second-order accuracy and maintains the linear form of the integrated equations. It is a satisfactory approximation for most problems (Wylie, 1993). The integration procedure leads to the equations

( ) ( ) +=Δ

+−+− CVVD

xVVapp APAPAP 02ρξρ (2.36)

( ) ( ) −=Δ

−−−− CVVD

xVVapp BPBPBP 02ρξρ (2.37)

where sub indices P, A, and B refer to the points in the characteristics grid. The numerical grid for solving the equations is shown in Fig. 2.3.

Fig. 2.3. Grid of characteristics.

24

If an unsteady friction model is used, the finite difference equations (2.36) and (2.37) look slightly different. An efficient approach incorporating the unsteady friction in MOC was demonstrated by Axworthy (2000).

2.1.6 Equations for oscillatory fluid transients When the transients are steady harmonic, the solution for Eqs. (2.7) and (2.18) can be obtained using the technique of separation of variables (Wylie, 1993). The complex pressure and volume flow rate are

xQgZxpp UCUD γργ sinhcosh −= (2.38)

xQg

xZp

gQ U

C

UD γ

ργ

ρcosh1sinh1

+= (2.39)

with ( )RLsCs +=γ the propagation constant, CsZC /γ= the characteristic impedance, ωσ is += the complex angular frequency, σ the decaying factor,

ω the angular frequency, 2/ agAC = the capacitance, gAL /ζ= the inertance, the factor ζ depends on the Reynolds number, R the resistance factor, and the subscripts U and D the upstream and downstream ends of the pipe, respectively. The quasi-steady resistance factor is

n

n

gDAQnR

2

1−=

ξ (2.40)

for turbulent flow in a circular pipe and Q denotes the mean volume flow rate, or

232

gADR ν= (2.41)

for laminar flow in a circular pipe, or

32

3gh

R ν= (2.42)

25

for laminar flow in a rectangular slit with 2h the height of the slit.

2.2 Pressure pulsation in one-dimensional tapered channel In Section 2.1.3, a special case for the equations of fluid transients was presented with friction neglected, which is the wave equation of acoustics (2.30). Pressure wave propagation in a tapered channel can be described with one-dimensional equations if the taper angle is small and the reflections from the channel walls can be omitted. The wave equation in a channel with non-constant cross-section can be obtained in a similar way as for a pipe or channel of constant cross-section by taking partial derivatives of Eqs. (2.17) and (2.28) with respect to t and x and combining the equations, which yields

0112

2

22

2=

∂∂

∂∂

+∂

∂−

∂

∂xp

xA

Atp

axp (2.43)

If the discussion is restricted to harmonic cases, t∂∂ / can be replaced with ωi , yielding the modified Helmholtz equation

0122

2=

∂∂

∂∂

++∂

∂xp

xA

Apk

xp (2.44)

in which ak /ω= is the wave number. In acoustics, the above equation (2.44) is also known as Webster’s equation (Morse, 1948).

2.3 Equations governing pulsation in three-dimensional case

The manifold-headbox system of a paper machine is a three-dimensional enclosure, in which pressure wave propagation cannot be described with one-dimensional equations. The harmonic pressure wave propagation is governed by the three-dimensional Helmholtz equation

022

2

2

2

2

2=+

∂

∂+

∂

∂+

∂

∂ pkz

py

px

p (2.45)

26

The modal pressure distribution in the rectangular enclosure (dimensions a × b × c) with rigid walls (corresponding to boundary condition 0/ =∂∂ np ) is

)/cos()/cos()/cos(),,( cznbymaxlAzyxp lmnlmn πππ= (2.46) where l,m,n correspond to the modes in each coordinate direction x, y, and z and A is the amplitude. The natural frequencies of the enclosure are obtained from

2222

2⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

cn

bm

alaflmn (2.47)

An analytical solution is obtained only for simple geometries such as the rectangular enclosure above. For complex geometries the solution is obtained with the finite element method (FEM). As was shown above, in the absence of friction we have a problem of acoustics in the approach piping and therefore we can use a FEM code of acoustics to solve pulsation problems. The commercial FEM code Abaqus (Abaqus Inc., 2006) is employed in this study.

2.4 Fluid-structure interaction In the theory presented above, the channel and pipe walls have been assumed rigid. In reality, pipelines and other elements in the approach flow system are flexible structures that affect the pressure wave propagation. In the conventional analysis of fluid transients the pipe elasticity is incorporated in the propagation speed of pressure waves (Wylie, 1993)

)/)(/(1

/eDEK

Ka+

=ρ (2.48)

where K is the bulk modulus of fluid, E the Young’s modulus of pipe material and e the pipe wall thickness. In the fluid-structure interaction (FSI) analysis the equations of the structure are added to the analysis and the structure and the fluid field are coupled at the interface. Since the 1970’s a significant amount of research has been done in the field of FSI, focused on understanding and quantifying the interactions between the transient flow and the vibration of the pipeline. This work has been

27

summarized in very comprehensive articles written by Tijsseling (1996) and Wiggert (2001). There are no general guidelines for when FSI should be taken into account except the time scale criterion discussed in Section 2.1.4. In 1989, Wylie (1989) said that 98 % of pipelines are not subjected to the significant influence of fluid movement and pressure wave traveling during the fluid transient, but since there is no convenient criterion for simple FSI inspection, it is necessary to perform FSI analysis for all pipelines. Casadei (2001) recommends FSI analysis always if the fluid is nearly incompressible or the structure is very deformable. During interaction, the pressure and the viscous stresses of the fluid act on the solid boundary and lead to structural deformations, which in turn affect the fluid flow and consequently the velocities, pressure, and viscous stresses of the fluid. Thus, the response of the system can only be determined if the coupled problem is solved. In the case of liquids, even a small structural deformation can have a significant effect (Papadakis, 2006). FSI concerns several types of waves that travel along the pipe and interact with each other. These are axial, flexural, rotational, radial, and torsional waves in the pipeline and pressure waves in the fluid. Three coupling mechanisms can be identified in FSI: Poisson coupling, friction coupling, and junction coupling (Wiggert, 2001). Poisson coupling is associated with the circumferential stress perturbations produced by liquid pressure transients that translate to axial stress perturbations by virtue of the Poisson ratio. The axial stress and strain perturbations travel as waves in the pipe wall at the speed of sound in pipeline material. Friction coupling is created by the liquid transient shear stresses acting on the pipe wall and it is usually insignificant compared to the other two coupling mechanisms. Junction coupling, which is often the most significant coupling mechanism, results from the reactions set up by unbalanced pressure forces and by changes in liquid momentum in discrete locations in piping such as in bends, tees, valves, and orifices. There are several methods for FSI coupling. In the uncoupled approach the fluid forces are first evaluated assuming the structure rigid. Next, the forces are transferred to the structural dynamics code, which calculates the structure displacement. This approach involves two separate analyses. It is obvious that this manner of proceeding introduces some approximations because the feedback influence of structural deformation on the fluid pressure is totally neglected. It is also possible to couple the codes at each iteration round (one-way coupling). Two-way coupling methods represent the state-of-the-art in FSI. In two-way coupling, FSI is defined with a mathematical model or two computer codes are coupled successively in such a way that the fluid code also

28

takes into account the structure deformations. Analysis can be performed using either acoustic-structural coupling or CFD structural coupling, depending on the case to be solved. In this study, acoustic-structural coupling is employed with the Abaqus code, which uses two-way coupling. The equation of acoustics that the Abaqus code solves (in the frequency domain) is the three-dimensional Helmholtz equation (2.45). The equation of dynamics of structures is in matrix notation

fuKuCuM =++ &&& (2.49) where M is the mass matrix, C the damping matrix, K the stiffness matrix, f the force vector, and u , u& , and u&& the displacement, velocity and acceleration vectors, respectively At the acoustic-structural interface the fluid and structure equations are coupled by the requirement that the fluid velocity normal to the structure must be equal to the normal component of the structural velocity

nn Vu =& (2.50) where nu& is the velocity of the structure. Furthermore, the acoustic boundary traction (acceleration) coupling fluid-to-solid reads as

nuT &2ω−= (2.51)

2.4.1 Equations of fluid-structure interaction in one dimension In studying one-dimensional phenomena, FSI in a pipeline is described by the system of 14 first-order partial differential equations. The equations (A.1)-(A.14) (in Appendix A) include 2 equations for the fluid, 2 for axial pipe motion, 8 for lateral motion, and 2 for torsional motion. The circumferential equations for the pipe are not important at low frequencies, i.e. the pipe diameter is much smaller than the wavelength (de Jong, 1994). The solution in the time domain is obtained using the method of characteristics (Tijsseling, 1996). In the frequency domain, the solution is obtained using the technique of separation of variables (Lesmez, 1990). Another possibility in the frequency domain is to use the transfer matrix model (de Jong, 1994) or the analytical model of Zhang (1999).

29

2.5 Equations for pulsating laminar flow in slit Equations governing highly viscous, pulsatile flow in a rectangular slit in the paper coating process are given in this section, both for the Newtonian and non-Newtonian fluid.

2.5.1 Newtonian flow The equation governing laminar, fully developed, pulsating flow with zero mean velocity in a rectangular slit (height 2h << slit width), shown in Fig. 2.4, for Newtonian fluid is obtained from the x-direction Navier-Stokes equation and is

2

2

yU

xp

tU

∂

∂+

∂∂

−=∂∂ ηρ (2.52)

where η is the dynamic viscosity of fluid (the kinematic viscosity ρην /= ). The harmonically pulsating pressure gradient is of the form

tixep

xp ω=∂∂ (2.53)

where xp is the pressure gradient amplitude. The equality in Eq. (2.53) holds only for the real part of the right-hand side.

Fig 2.4. Geometry of slit.

30

The analytical solution of Eq. (2.53) is obtained as

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+−= yiCyiC

ip

yU xνω

νω

ρωcoshsinh)( 21 (2.54)

where C1 and C2 are unknown constants. The boundary conditions of the problem are zero flow velocity at the wall and the velocity profile is symmetric with respect to the x-axis

⎪⎩

⎪⎨⎧

=∂

∂=

0)0(0)(

yU

hU (2.55)

The final solution for the velocity profile with the boundary conditions is

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

= 1sinh

cosh)(

hi

yi

ip

yU x

νω

νω

ωρ (2.56)

The mean velocity is

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛== ∫ 1tanh1)(1

0

hiih

ipdyyU

hV x

h

νω

ων

ωρ (2.57)

and the volume flow rate per unit width is obtained from

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛==′ hhi

iip

VhQ xνω

ων

ωρtanh

22 (2.58)

2.5.2 Non-Newtonian flow For non-Newtonian liquids the shear stress is not a linear function of constant viscosity and the velocity gradient as yU ∂∂= /ητ . For fluids obeying the

31

power-law, the viscosity depends on the shear rate yU ∂∂= /γ& of the fluid and is

1−= nkγη & (2.59) where k is the consistency index and n the power-law index. The shear stress is now

γητ &= (2.60)

The equation governing non-Newtonian steady flow in a slit is

n

kh

xp

yU /1

⎟⎠⎞

⎜⎝⎛∂∂

−=∂∂ (2.61)

Eq. (2.61) has an analytical solution which is (Tadmor, 1979)

( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛∂∂

+=

+ nnn

hy

kh

xp

nnhyU

/1/11

1)( (2.62)

The volume flow rate per unit width through the slit is

n

kh

xp

nnhQ

/12

122

⎟⎠⎞

⎜⎝⎛∂∂

+=′ (2.63)

Eq. (2.52) governing the pulsating laminar flow now becomes

n

yU

yk

xp

tU

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+∂∂

−=∂∂ρ (2.64)

An analytical solution of Eq. (2.64) does not exist and it can be solved numerically, e.g. using the CFD code FLUENT (Fluent Inc., 2005), which is based on the control volume method.

32

2.6 Solution of equations governing pulsation In rigid 1D cases the solution is obtained by solving the equations of harmonic fluid transients. In 3D rigid cases as well as in FSI cases a commercial FEM code Abaqus is employed. The fluid equation solved is the 3D Helmholtz equation (2.45). The shape of elements is hexahedral whenever possible. With some difficult shapes, a part of the structure must be modeled with tetrahedral elements. The fluid elements used in Abaqus code are AC3D8 and AC3D4. The structures are modeled with shell elements S4R and S3. The wave equation governing the vibration of a plate is

( ) 0132

2

2

24 =

∂

∂−+∇

tz

Ehz νρ (2.65)

where z is the out-of-plane displacement, E the Young’s modulus, 2h the thickness of plate, and ν the Poisson constant. The equations governing the vibration of shells in Abaqus are described by the classical (Kirchhoff) shell theory (Budiansky, 1963). The pressure results from the acoustical solution are interpolated and applied as pressure boundary conditions on the structural elements. Displacements from the structural analysis are interpolated and applied to the boundary of the acoustical mesh. The interaction between acoustic and structural media in the Abaqus code is modeled using the surface-based interaction approach. In this approach the acoustic and structural meshes can have different node numbering and the meshes at the interface need not be spatially coincident. The acoustical side receives point tractions or fluxes based on interpolation with the shape functions from the structural side. At the start of an analysis, the projections of acoustical nodes onto the structure surface are found and the areas and normals associated with the acoustical nodes are computed. The projections are points on the structural surface and the structural nodes in the vicinity of this projection are identified. Variables at the acoustical nodes are then interpolated from variables at the identified structural surface nodes near the projection. The values on the acoustical surface are constrained to equal values interpolated from the structural surface. The (time-averaged) Navier-Stokes equations and the continuity equation give the full description of the (turbulent) flow field. At low flow velocities that correspond to the Mach number 1.0< , the mean flow does not affect the low

33

frequency pressure waves. In the case of harmonic fluid transients, these equations are too time-consuming to solve with the finite volume method. A more feasible approach is to solve the equations of fluid transients or acoustics. A comparison of CFD and acoustical calculations in a nuclear power plant pressure transient case can be found in Lestinen (2006). The results from these two approaches were similar. The only way to calculate pulsation in the paper coating process for non-Newtonian fluid is to solve it with a CFD code. In this study the solution is obtained with the Fluent code (Fluent, Inc., 2005).

34

3 PULSATION IN TAPERED CHANNELS The slice channel of the headbox is the last component of the paper machine piping system where the fluid is accelerated to machine speed. The effect of this tapered channel on pulsation is analyzed in this chapter. There are other functions besides that of the paper machine headbox in which the flow or pressure waves in a tapered channel are important. These include e.g. the flow of blood in arteries (Belardinelli, 1992) and the flow in human lungs (Grotberg, 1984). In physiological flows, taking the fluid-structure interaction into account is very important. Other relevant examples include musical instruments (Arenas, 2001; Myers, 2004) and the flow in the inkjet (Shin, 2005). There have been some analyses done for the transients in tapered sections. Adamkowski (2003) compared different formulations in MOC for calculating pressure transients in an expanding or contracting section for a valve closure at the downstream end of a pipe. Tahmeen (2001) developed a method for simulating transient responses of tapered fluid lines by considering the frequency-dependent effects of viscosity.

3.1 Rigid, one-dimensional model of pipeline The procedure for modeling pressure pulsation in a rigid, one-dimensional piping system is described in this section.

3.1.1 Pipe/channel sections The approach flow system of a paper machine consists of a complex piping system with several components, which is presented in Fig. 1.2. The system also includes several pipe bends. If the system is rigid, the bends do not affect the wave propagation at frequencies at which only plane waves propagate (Fahy, 2001). In order to solve the pressure and volume flow rate oscillation using the equations of fluid transients for harmonic excitation in a pipe or channel with non-constant cross-sectional area, the pipe/channel is divided into sections of constant cross-section, shown in Fig. 3.1. At the junction of two sections the pressures on both sides of the junction are equal and the continuity condition holds.

35

Fig. 3.1. Dividing linearly tapered channel into sections of constant cross-section. The pressure and volume flow rate oscillation can be determined at each point of the pipe/channel with the use of Eqs. (2.38) and (2.39) and the boundary conditions. The equations for one pipe/channel section are

lQ

gl

Zp

gQ

lZgQlpp

iDiC

DU

iCDiDU

i

i

i

i

iiii

11

11

cosh1sinh1

sinhcosh

1

1

1

1

1111

++

++

+

+

+

+

++++

+=

+=

γρ

γρ

γργ

1+

=ii UD HH (3.1)

1+

=ii UD QQ

where l is the length of channel section.

3.1.2 Cross-section changes in piping and headbox There are several contractions and expansions between the manifold and the slice channel in the paper machine headbox. Also the modeling of tapered channel with straight channel sections involves cross-section changes. In acoustics, the pressure wave is divided into incident, reflected, and transmitted waves (Davies, 1988). In the approach of harmonic fluid transients the total pressure is used. The different impedance of pipe sections takes into account reflection and transmission and no transfer matrix is needed for the cross-section change. At low flow speeds the velocity head, i.e. the change in kinetic energy, may also be neglected at the junction.

3.1.3 Assembly of different components To construct a model describing the entire piping system, the equations can be cast in matrix form. The matrices Fi for all the elements are multiplied together and a relationship between the quantities at upstream and downstream ends of the system is obtained as

36

U

NND Q

pQp

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡− 121... FFFF (3.2)

The boundary conditions are introduced in the upstream and downstream vectors. If there are several exciters or the same source excites the system at different frequencies, as is the case for pumps and screens, each excitation frequency or source is treated separately and the total response is obtained by summing all the single frequency responses.

3.2 Analysis of tapered slice channel The pressure and volume flow rate oscillation in slice channels of different shapes were compared. The channel walls were assumed rigid. Analytical solutions are derived for linearly and exponentially tapered channels. For other channel shapes the Webster’s equation (2.44) was solved using the equations of harmonic fluid transients for non-uniform channel sections shown above. The boundary conditions, the channel height at upstream and downstream ends, and the channel length were the same in all cases. For a linear channel the effect of different parameters such as the channel dimensions and the wave speed on the volume flow rate oscillation was examined. The shapes of different channels are illustrated in Fig. 3.2. The comparison between different geometries was performed using typical values for the slice channel of a paper machine. The channel dimensions were y1 = 0.05 m (inlet), y2 = 0.01 m (outlet), L = 1 m, the oscillation frequency f = 60 Hz, corresponding to the blade passing frequency of the headbox feed pump, and the wave speed a = 1200 m/s for water (ρ = 1000 kg/m3). The pressure oscillation amplitude at the upstream end was Δp = 1000 Pa.

37

Fig. 3.2. Channel height of different channel shapes.

3.3 Analytical solutions of Webster’s equation Analytical solutions of Webster’s equation (2.44) can be found for linearly and exponentially tapered channels.

3.3.1 Linearly tapered channel The equation for the cross-sectional area of a linearly tapered channel and constant width (see Fig. 3.2) can be written

( )xmbbxyxA +=+= )0()( (3.3) in which [ ] LyLyb /)0()( −= , bym /)0(= with y(0) the channel height at the upstream end, y(L) the height at the downstream end, and L the channel length. When Eq. (3.3) is substituted into Eq. (2.44), it yields

0122

2=

∂∂

+++

∂

∂xp

xmpk

xp (3.4)

The analytical solution of Eq. (3.4) derived by the author is

38

( )[ ] ( )[ ]mxkYCmxkJCxp +++= 0201)( (3.5)

in which J0 and Y0 are the Bessel functions of first and second kind and C1 and C2 unknown constants to be determined from boundary conditions. The boundary conditions in the paper machine problem are the known pressure amplitude pU at the upstream end and the zero pressure at the downstream end representing the discharge to the atmosphere.

⎩⎨⎧

==

0)()0(

Lppp U (3.6)

The final solution with the boundary conditions (3.6) is

( )[ ] ( )[ ] ( )[ ] ( )[ ]

( )[ ] ( ) ( ) ( )[ ]LmkYkmJkmYLmkJxmkYLmkJLmkYxmkJ

pxp U +−+++−++

−=0000

0000)( (3.7)

The velocity oscillation (in acoustics called the particle velocity) is obtained from

xpiV∂∂

=ρω

(3.8)

The volume flow rate oscillation is obtained from

( ) )(xVAxQ = (3.9)

3.3.2 Exponentially tapered channel If the shape of the channel is exponential (see Fig. 3.2), the equation of cross-sectional area is ( ) hxeyxA /0)( = , where [ ])0(/)(ln/ yLyLh = . Webster’s equation (2.44) becomes now

0122

2=

∂∂

++∂

∂xp

hpk

xp (3.10)

and the general solution of Eq. (3.10) derived by the author is

39

( )[ ] ( )[ ]2/1222/122 141

22

1412

1)(−−−−−

+=kh

hx-kh

hx

eCeCxp (3.11) Applying the same boundary conditions (3.6) as in the case of linear channel yields the solution

⎥⎥⎦

⎤

⎢⎢⎣

⎡+−

⎥⎥⎦

⎤

⎢⎢⎣

⎡+−

⎥⎥⎦

⎤

⎢⎢⎣

⎡++−−

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−−

+−

−=

21

2221

22

21

2221

2221

2221

22

41121411

21

4141214141

21

)()kh-(L

h)kh-(L

h

)kh-x(x)kh-L(Lh

)kh-x(x)kh-L(-Lh

U

ee

eepxp

(3.12)

3.3.3 Other channel shapes If the channel shape is parabolic or e.g. a 3rd degree curve, an analytical solution of Eq. (2.44) cannot be found (Campos, 1984) except for a shape called catenoidal, and the equation must be solved numerically. Numerical solution can be obtained using the approach of harmonic fluid transients explained in Section 3.1.

3.4 Comparison of different channel shapes The results of pressure and volume flow rate oscillation in channels of different shape are presented below. Figs. 3.3 and 3.4 show the comparisons for different channel geometries. For the linear and exponential channels, analytical solutions (3.7) and (3.12) derived above are used. For other channel shapes, Eq. (2.44) was solved numerically. Analytical and numerical results were exactly the same for cases for which an analytical solution was available. In Fig. 3.3 the pressure oscillation along the channel length is presented for different geometries. The pressure amplitude decreases at about the same rate for all channel shapes and not much can be deduced from this figure concerning the velocity or volume flow rate pulsation. In Fig. 3.4, we can see the volume flow rate oscillation, which increases towards the channel outlet. An interesting fact in Fig. 3.5 is that the magnitude of oscillation is highest for the linear channel. The volume flow rate oscillation is lowest for the parabolic channel, being approximately 28 % lower than in the linear channel.

40

Fig. 3.3. Pressure oscillation in slice channel normalized with inlet pressure amplitude pU for different channel shapes.

Fig. 3.4. Volume flow rate oscillation amplitude in linear channel normalized with Q at upstream end. The exponential and 3rd degree curve channels are placed between linear and parabolic channels in volume flow rate oscillation, the latter having a higher oscillation, only about 10 % less than in the case of linear channel.

41

Fig. 3.5. Volume flow rate oscillation amplitude in different channels normalized with results of linear channel.

Fig. 3.6. Volume flow rate oscillation in linear channel for exact and approximate solution of equations of fluid transient using different number of sections.

42

The magnitude of the oscillation amplitude of volume flow rate is quite different in channels of different shapes already at the upstream end of the channel, as shown in Fig 3.5. The volume flow rate oscillation is a system-dependent quantity. It is strongly related to the kind of system there is before and after the point of interest and to what the boundary conditions are. An optimization procedure could be employed to find a shape that gives lower volume flow rate oscillation amplitude than the parabolic shape.

3.5 Comparison of analytical and numerical solutions It is shown in Fig. 3.6 that the analytical and numerical solutions give identical results. Fig. 3.6 shows the volume flow rate oscillation in a linear channel along the channel length, calculated using the approach of harmonic fluid transients and compared with the analytical solution of Eq. (3.7). The solid line is for the analytical solution and the dots for numerical solutions obtained using 2, 5, 10, 50, and 100 sections.

If very few sections are used, the approach of fluid transients gives too small values for the volume flow rate oscillation, but as the number of segments is increased, the solution converges quickly to that of the exact solution. With 50 sections the error is negligible compared to the exact solution. Also, the method of characteristics gives results identical to those above.

3.6 Effect of channel dimensions on volume flow rate oscillation in linear channel

The effect of channel dimensions on the volume flow rate oscillation in a linear channel on the basis of analytical result (3.7) is illustrated in Fig. 3.7. Only one quantity was varied at a time, while other parameters were kept constant in each case. These constant values are given at the beginning of this chapter. The results are presented at the downstream end. It is not meaningful to study the effect of pulsation frequency on the volume flow rate oscillation. If there is a long pipe or channel before the tapered section, the results will change considerably because the natural frequencies of the system depend on the whole length of the piping system. Nevertheless, the results presented in this section have the same characteristics also at frequencies other than 60 Hz, and also if the tapered channel is connected to a long channel upstream of it.

43

Fig. 3.7. Volume flow rate oscillation at outlet as function of contraction ratio normalized with contraction ratio 5. Making the channel shorter and thus increasing the taper angle, or the contraction ratio y1/y2, increases the volume flow rate oscillation, which can be seen in Fig. 3.7. On the other hand, a high contraction ratio in the slice channel has been found favorable for fiber orientation in paper (Paulapuro, 2000). Variation in channel dimensions is presented above only for the linear channel, but also other channel shapes yield behavior similar to that in the linear case.

3.7 Effect of air on pulsation A major factor that affects wave speed propagation in liquid is the free air content in it. Free air is always present in paper stock (Paulapuro, 2000). The fluid may be accelerated in a tapered channel to a velocity of approximately 20 m/s. The jet velocity is generated with a pressure difference between upstream and downstream ends. The higher upstream pressure compresses free air and its volume is greatly reduced compared to that at the downstream end. The bulk modulus is now obtained from (Wylie, 1993)

44

⎟⎟⎠

⎞⎜⎜⎝

⎛−+

=

11abs

liq

abs

a

liq

pK

pRTm

KK (3.13)

with ma the mass of free air per cubic unit of volume, R the gas constant, T the temperature, Kliq the compressibility of liquid, and pabs the absolute pressure. A typical 200 kPa pressure difference for the slice channel was used and the non-constant bulk modulus resulting from this difference was calculated from Eq. (3.13). The volume flow rate oscillation amplitude was compared to a case with a constant 1000 m/s wave speed. With the above pressure difference, the wave speed range calculated from Eq. (2.14) in the linear channel, the air content being 0.01 %, is approximately from 1300 m/s at the upstream end to 870 m/s at the downstream end. Non-constant wave speed is easy to incorporate in the numerical solution of harmonic pressure transients, with each channel section having a different wave speed and thus different characteristic impedance.

Fig. 3.8. Volume flow rate oscillation amplitude in linearly tapered channel.

45

In Fig. 3.8 it can be seen that the change in wave speed resulting from the absolute static pressure drop affects the volume flow rate very little, leading only to a difference of approximately 1 % at the outlet. Thus, a constant wave speed assumption can be used.

3.8 Effect of vanes on pulsation The guide vanes in the slice channel, shown in Fig. 1.4, are used in some headboxes. They have several purposes. In some applications they are needed to generate turbulence and mixing by surface friction and flow separation from the trailing edge. They are also used for controlling the turbulence length scales inside the slice chamber. In multi-layer paper production the vanes separate layers of different stock until the layers are brought together just before the slice channel exit. The vanes are usually slightly shorter than the slice channel and they are simply supported at the upstream end. The thickness of vanes is a few millimeters. Pulsation coming from the upstream excites the vanes and causes them to vibrate. Some natural frequencies of the vanes may coincide with the excitation frequency range and thus there is a possibility for high amplitude vibration.

Fig. 3.9 Velocity oscillation magnitude and vane vibration shape in tapered channel with vane at 40 Hz. Vane deformation is greatly exaggerated.

46

A tapered channel with the same dimensions as in Section 3.4 was analyzed with a vane of 3 mm thickness inside the channel (ρ = 7800 kg/m3, E = 205 GPa, ν = 0.29). The vane was hinged (lateral and transverse displacement zero) at the upstream end and the pressure pulsation amplitude at the upstream end was 1000 Pa. The channel walls had a thickness of 10 mm (same material as vane). The solution was obtained with the Abaqus code. The fluid-structure interaction analysis shows that the vane vibration affects the volume flow rate oscillation at the channel outlet very little, the greatest difference, 10 %, occurring if the channel walls are assumed rigid, and no difference at all occurring if flexible walls are included in the FSI analysis. Even if the excitation occurs at the natural frequency of the vane, the amplitude of displacement of the vane is quite moderate, some dozens of micrometers. Another phenomenon caused by vanes is the vortices shed from the vanes. The vibration caused by vortices shed from the vane tip modifies the flow field considerably, which was shown by Eloranta (2005).

47

4 PULSATION IN PAPER MACHINE APPROACH PIPING AND HEADBOX

Above we have discussed pulsation phenomena in short tapered sections. In this chapter, the approach piping system of a paper machine and all the elements of the headbox are taken into account in the analysis. Rigid, 1D, 3D, and FSI models are compared. The real paper machine model simulations are compared with the experimental findings.

4.1 Paper machine measurements An abundance of measurement data from real and pilot paper machines is available (Metso Paper, Inc., 2006). To study basis weight variation, several pressure sensors are placed in the approach flow system and headbox. Paper samples are taken simultaneously and the variation in machine-direction basis weight is measured using beta radiation absorption. The pressure peaks found in the approach flow system and headbox are compared with those found in the basis weight variation spectrum. Even though there is a lot of measurement data available, the data has not been collected for validating or performing a pulsation analysis. The exact geometry of the approach piping system should be known. The piping and the structural supports need to be known and also the material thickness. Also the value of the wave speed should be available (which can be easily determined if the pressure signal data is available). The exact location of the pressure sensors should be known as well as the detailed geometry of the manifold and the headbox. The measurement data has been collected from paper machines around the world over a period of several years and the systems in which the measurements have been taken may no longer exist in the same form. To obtain such data as that mentioned above seems an impossible task. Therefore, no precise comparison of the calculations can be made. It is possible, however, to model the system and to examine, if the modeled and measured results are of the same magnitude and if the modeled results bear the same features as the measurement data.

48

4.2 Volume flow rate oscillation in long rigid pipeline It is not possible to evaluate reliably the magnitude of velocity or volume flow rate oscillation on the basis of the pressure measurement data from one pressure sensor. The only case when this is possible is when the pipe is long and can be modeled as an infinitely long pipeline, which means that there are no reflections from the pipe terminations. The corresponding downstream boundary condition is the impedance ( ) CDDD ZQgpZ −== ρ/ . For the volume flow rate oscillation at the upstream end of an infinite pipeline with the known pressure oscillation Up , Eqs. (2.38) and (2.39) yield the expression

( )( )

l

C

U

C

UU e

gZp

llZgllp

Q γ

ργγργγ 2

coshsinhcoshsinh

−=−

+= (4.1)

The modulus of the exponential term equals 1 in Eq. (4.1) and thus the magnitude of volume flow rate oscillation depends only on the pressure oscillation and the characteristic impedance, yielding in the frictionless case

aAp

Zp

Q U

C

UU ρ

=−= (4.2)

4.3 Rigid vs. FSI solution in straight pipe To obtain an idea of how the FSI solution differs from the results of a rigid model we shall start with a simple model of a straight rectangular steel channel. The channel length is 10 m, the channel cross-section is 0.1 x 0.1 m, the wall thickness 5 mm, and the wave speed 1000 m/s. The material properties for all the structures in the study are the density 7800 kg/m3, the Young’s modulus 205 GPa, and the Poisson constant 0.29. The material damping was not included. The channel is simply supported (displacement in all coordinate directions zero) at both ends. The harmonic pressure oscillation amplitude of 1000 Pa is imposed at the upstream end and the channel discharges to the atmosphere (p = 0). The solution is obtained for the rigid case from Eqs. (2.38) and (2.39) and using the approach described in Section 3.1. For the FSI case the solution is obtained using the Abaqus code. The comparison of velocity oscillation amplitude along the channel length is shown in Fig. 4.1. The velocity oscillation amplitude at the channel outlet over the frequency range of 10-100 Hz is illustrated in Fig. 4.2.

49

10 Hz 20 Hz

40 Hz

60 Hz

⎯ rigid solution --- FSI solution

Fig. 4.1. Velocity oscillation amplitude along channel length. The FSI model gives higher velocity oscillation at the channel outlet than the rigid model at most of the frequencies analyzed and there is quite a large difference in the magnitude according to Figs. 4.1 and 4.2. Rigid calculations exhibit strong resonance, as seen in Fig. 4.2, whereas the more realistic FSI calculations do not. FSI changes the frequency spectrum obtained with the rigid model and tends to separate coinciding fluid and structural natural frequencies. The same phenomenon has been observed by other researchers (Wiggert, 2001).

50

Fig. 4.2. Velocity oscillation amplitude at channel outlet as function of frequency. Now, let us compare the velocity oscillation in an infinitely long channel and the 10 m long channel above. Eq. (4.2) gives a velocity oscillation amplitude of 0.001 m/s in the infinitely long pipe. The result is of the same magnitude as in Fig. 4.1. It must be noted that in a pipeline with reflections the velocity oscillation depends on the excitation frequency.

4.4 1D rigid model equations for paper machine system The equations for modeling the rigid 1D piping system and headbox with the equations of harmonic fluid transients are presented below. The piping system components are divided into k sections. The transfer matrix for the straight pipe is

1

2

coshsinh1sinhcosh

coshsinh1sinhcosh

...coshsinh1sinhcosh

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−

−

⋅⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−

−

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−

−=

llZ

lZl

llZ

lZl

llZ

lZl

C

C

C

C

kC

C

p

γγ

γγ

γγ

γγ

γγ

γγF

(4.3)

51

and similarly we obtain the transfer matrix for the manifold Fm and the tapered channel Ft. The transfer matrix in the manifold is calculated until the middle of it and then the calculation path turns to the tube bunch. The lumped parameter approximation (Wylie, 1993), which is valid for short pipes, is used to model the fluid behavior in the tube bunch, the equalization chamber, and the turbulence generator, and the transfer matrix for those elements is

⎥⎦

⎤⎢⎣

⎡ −=

101 Lli

ic

ωF (4.4)

where L is the inertance of the component cross-section and l the length. The equation relating the upstream and downstream pressure and volume flow rate is

Upmccct

D Qp

Qp

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡FFFFFF

123 (4.5)

The pressures at the upstream and downstream ends are known from boundary conditions and the unknown volume flow rates can be solved from the above equation (4.6). Next, the pressure and the volume flow rate can be calculated at each point of the system.

Fig. 4.3. 3D rigid paper machine model geometry. Division into different components and sections for computation with 1D model.

52

4.5 Laboratory measurements and analysis of tapered channel – manifold system

A manifold-headbox-type system was built in a laboratory environment. The purpose was to measure the harmonic velocity oscillation amplitude at the outlet of the channel using particle image velocimetry (PIV). The pulsation was created by normal operation of a centrifugal pump. The tapered channel system used in the measurements is shown in Fig. 4.10. The channel height was 0.05 m at the upstream end and 0.01 m at the downstream end (outlet). The structural thickness was 5 mm and some stiffeners were included in the structure. The velocity (oscillation) was measured from the channel outlet using the PIV at four different locations (diamonds in Fig. 4.4). The excitation was created by a centrifugal pump which had 3 vanes and was running at 1200 rpm. Therefore, pulsation occurs at 20 Hz and its multiples. The channel was discharging to the atmosphere. The same system was also analyzed with the 3D FSI model.

Fig. 4.4. Geometry of tapered channel used in measurements: ♦ velocity measurement locations x = 0.05, 0.3, 0.7, 0.95 m; • pressure measurement.

53

Table 4.1. Velocity oscillation amplitude measured with PIV at channel outlet; excitation frequencies 40, 60, and 80 Hz.

x [m] f [Hz] 0.05 0.3 0.7 0.95

40 - - 1.7E-03 - 60 1.2E-03 1.5E-03 - 1.1E-03 V [m/s] 80 - - 1.3E-03 -

Fig. 4.5. Measured pressure spectrum (y-axis voltage) in tapered channel, location indicated by ● in Fig. 4.4. Table 4.1 shows the measured velocity oscillation amplitudes at the tapered channel outlet. We can see that at 60 Hz the velocity oscillation is found at three measurement points, but at one point the harmonic velocity oscillation could not be distinguished from the noise of the signal. If we look at the pressure spectrum measured in the channel in Fig. 4.5, we can see that the peak of 60 Hz is quite low compared to the other frequencies. At the location where the 60 Hz component was not found in Table 4.1 we can see velocity oscillation at 40 and 80 Hz of the same magnitude as for 60 Hz at other locations.

54

Fig. 4.6. Simulated pressure oscillation amplitude in tapered channel-manifold system. In Figs. 4.6 and 4.7 we can see modeling results for the system used in the experiments. The pressure oscillation amplitude at the excitation frequency of 40 Hz is illustrated in Fig. 4.6. In Fig. 4.7 the velocity oscillation amplitude is plotted at the tapered channel outlet at excitation frequencies of 40, 60, and 80 Hz. The velocity oscillation has a high peak at the location where there was a peak found in the measurements at 40 and 80 Hz, i.e. at x = 0.7 m. At the frequency of 60 Hz the velocity oscillation amplitude is of the same magnitude at all the measurement points. The computation results show that there would be velocity oscillation of the same magnitude at x = 0.7 m as at the other points, which is not found in Table 4.1. Otherwise the measurements and the experiments match very well. The explanation for the velocity oscillation at 40 and 80 Hz at x = 0.7 m can be found by comparing Table 4.1 and Fig. 4.7. Measurements also show that there is variation in the amplitude of velocity oscillation in the cross-direction. The modeling and measurement results show that the FSI model must be employed when performing a pulsation analysis in large flexible structures such as the paper machine manifold and headbox. The features found in the rigid model calculations above cannot be confirmed with the measurements.

55

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

x [m]

V* 40 Hz

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

x [m]

V* 60 Hz

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

x [m]

V* 80 Hz

Fig. 4.7. Simulated velocity (normalized with value at x=0.0) oscillation amplitude at tapered channel outlet; excitation frequencies 40, 60, and 80 Hz.

56

4.6 Real paper machine model, headbox width 1 m Approach piping and a headbox similar to those in a real paper machine were modeled to study how well the results agree with measurements. The geometry of the model is shown in Fig. 4.8. The approach pipe length is approximately 15 m and the pipe diameter 0.3 m. The headbox width is 1 m, the slice channel length is 0.8 m, the height at the upstream end 0.1 m, and that at the downstream end 0.01 m. Between the manifold and the slice channel, there are the manifold tube bunch, the equalization chamber, and the turbulence generator. In the real paper machine there is also a recirculation outlet to which approximately 10 % of the flow is directed. Including the recirculation outlet in the analysis affects the results very little. This will be shown later and is therefore excluded from the analysis. The dilution system shown in Fig. 1.4 is also excluded. The volume flow rate coming from the dilution system inlet is small compared to the volume flow rate in the approach piping. Also, pulsation coming from the dilution system is in most cases at a lower level than pulsation originating from the headbox feed pump or the screens. Hence, excluding this source of pulsation does not much affect the pulsation phenomena that can be seen in the headbox.

Fig. 4.8. Model of approach piping and headbox in realistic paper machine. Pressure sensor location indicated with •.

57

The width of a real paper machine in production use can be from a few to over ten meters. The paper machine of 1 m width is modeled because the validation of results using a computational model of this size is easiest. This is because the pilot paper machines used to make trials and test new developments are of that magnitude. The validation measurements are much more easily conducted with these machines than with the paper machines that are in production use in paper mills. Also, since there are paper machines sold with the width between 1 and 10 meters, we can see the effect of the machine width on the computational results. Furthermore, it is easier to do computations and different variations in computational models with the machine of 1 m width. A machine of 10 m width needs approximately 10 times more elements and the computational cost is much greter if we want to achieve the same accuracy in the analysis. Therefore, most of the computational cases are done using the machine of 1 m width, but cases exist in which a width of 10 m has been used Pressure pulsation was analyzed in the system using several different parameters. The pressure pulsation amplitude at the upstream end of the piping, corresponding to the pressure amplitude after a screen or a pump, was 1000 Pa at all frequencies and in all of the cases. The different computational cases are shown in Table 4.2. The computational mesh and the structural supports in the piping and the headbox are shown in Fig. 4.9 and they are similar in all the cases, unless otherwise mentioned. The symbol e is the structure thickness.

Fig. 4.9. Mesh and structural supports (displacement in all coordinate directions zero, nodes shown in red) for 1 m paper machine model.

58

Table 4.2. Parameters for different paper machine models.

Case 1 rigid structure, a = 1000 m/s Case 2 FSI, e = 5 mm, a = 1000 m/s Case 3 FSI, e = 10 mm, a = 1000 m/s Case 4 FSI, e = 5 mm, a = 1000 m/s, headbox rigidly supported Case 5 FSI, e = 5 mm, a = 500 m/s

Fig. 4.10. Pressure pulsation amplitude for rigid case (case 1), 40 Hz. In Fig. 4.10 the pressure pulsation amplitude in the headbox can be seen for the rigid case 1. The pressure loss elements and the changes in cross-section effectively even out the pressure profile. The 1D (transfer matrix model) and 3D (Abaqus) computations are compared in Fig. 4.11. The 1D model equations for the rigid case were shown in Section 4.4. For the 3D model, the results are shown at the centerline of the headbox outlet. The 1D and 3D computations give very similar results in the real rigid paper machine case. There are greater differences close to the resonance frequencies. In conclusion, for a rigid paper machine system with a small headbox width, the 1D model is sufficient in conduction a pulsation analysis. The pressure pulsation amplitude in the first FSI case (case 2) is seen in Fig. 4.12. There can be pressure variations in the longitudinal and also in the cross-direction.

59

0,00E+00

1,00E-02

2,00E-02

3,00E-02

4,00E-02

5,00E-02

6,00E-02

0 20 40 60 80 100 120

f [Hz]

V [m

/s] 3D1D

Fig. 4.11. Comparison of velocity oscillation amplitude at headbox outlet for 1D and 3D solutions for rigid case 1.

Fig. 4.12. Pressure pulsation amplitude in headbox, FSI model case 2, 50 Hz. The velocity oscillation amplitude is compared at the headbox outlet between cases 1 and 2 in Fig. 4.13.

60

In Fig 4.13 it can be seen that the velocity oscillation amplitude is greater in the rigid case than when FSI is taken into account. Only at two frequencies, which are 10 and 30 Hz, does the FSI model give larger oscillation amplitude than the rigid model. If we assume the machine to run at 20 m/s, only at those frequencies would there be excess basis weight variation (1.6 and 0.4 %) in paper. These frequencies are related to local natural frequencies of the headbox plates. The deformed shape of the structure is shown in Fig. 4.14 at 10 Hz. The very high amplitude vibration in headbox plates is the reason for high velocity oscillation amplitude at the outlet. Fig. 4.15 shows the pressure oscillation amplitude along the system length for the FSI case 2 for frequencies 10 and 50 Hz. In the former case, the pressure pulsation amplitude is at a very high level near the system outlet and it produces high velocity oscillation, as can be seen in Fig. 4.16. In the latter case, the pressure pulsation amplitude is quite low from 8 meters onwards until the system outlet and the resulting velocity oscillation at the outlet is considerably lower. The results suggest that low pressure pulsation amplitude is required in the proximity of the system outlet to achieve low velocity variation. Next we shall look at the pressure values shown in Table 4.3 at typical pressure measurement locations in the manifold, which are seen in Fig. 4.8, i.e. “1” the manifold inlet side and “2” the other extremity. We can see in Table 4.3 the same that can be seen in Fig. 4.15. The pressure pulsation amplitude is very dissimilar at different frequencies. The high pressure amplitude indicates very probably high velocity oscillation amplitude at the system outlet, but no direct conclusions can be drawn from these pressure values regarding what the velocity or basis weight variation could be. For instance, at the frequency of 30 Hz the pressure pulsation amplitude is approximately twice as high as at 10 Hz. However, the velocity oscillation at 10 Hz is over three times higher than at 30 Hz, which can be seen in Fig. 4.13, and the very high pulsation amplitude measured at 90 Hz does not cause significant velocity oscillation at the system outlet and thus basis variation at that frequency. Also, the velocity oscillation at frequencies of 50 and 90 Hz are of the same magnitude, but the pressure pulsation amplitude in the manifold is at the latter frequency considerably higher.

61

0,00E+00

5,00E-02

1,00E-01

1,50E-01

2,00E-01

2,50E-01

3,00E-01

3,50E-01

0 20 40 60 80 100 120

f [Hz]

V [m

/s] FSIRigid

0,00E+00

1,00E-03

2,00E-03

3,00E-03

4,00E-03

5,00E-03

6,00E-03

7,00E-03

8,00E-03

0 20 40 60 80 100 120

f [Hz]

V [m

/s] FSIRigid

Fig. 4.13. Velocity oscillation amplitude at headbox outlet, case 1 and 2, larger and smaller scale. This suggests that if the pressure pulsation amplitude measured in the manifold at two frequencies are equal, there will be greater velocity oscillation at the outlet and thus basis weight variation at the lower frequency. This is better illustrated in Fig. 4.16, where the quantity “velocity oscillation amplitude at headbox outlet” divided by “pressure oscillation in manifold location 1” is plotted against the frequency. The phenomenon discovered cannot be generalized to all systems because there may be local natural frequencies of structures that change the case at some frequencies or fluid natural frequencies and modes, which is the case for

62

the frequency of 30 Hz. However, the general trend is that the higher frequency pulsation is less severe than the lower one. This phenomenon has been found in the paper machine measurements with similar ambiguities as the frequency of 30 Hz. In Section 1.3 fourdrinier machines were said to exhibit strong basis weight variation at the frequency range of 10-20 Hz. According to the computations, this phenomenon is also possible in gap former machines, assuming that the ratio “pulsation amplitude - mean value” is equal at the headbox outlet and in the basis weight of paper.

Fig. 4.14. Deformed shape (greatly exaggerated) of structure and pressure pulsation amplitude for case 2 at 10 Hz excitation.

63

0.00E+00

5.00E+02

1.00E+03

1.50E+03

2.00E+03

2.50E+03

3.00E+03

3.50E+03

4.00E+03

0 5 10 15 20

x [m]

p [P

a]

10 Hz

0.00E+00

2.00E+02

4.00E+02

6.00E+02

8.00E+02

1.00E+03

1.20E+03

1.40E+03

1.60E+03

1.80E+03

2.00E+03

0 5 10 15 20

x [m]

p [P

a]

50 Hz

Fig. 4.15. Pressure pulsation amplitude along system length, FSI case 2. The high pressure pulsation amplitude measured far from the system outlet does not necessarily mean that there is strong velocity and basis weight variation. If high pressure pulsation is found close to the system outlet, the pulsation of that frequency is likely to cause high velocity oscillation. However, it cannot be said how strong the oscillation will be.

64

Table 4.3. Pressure pulsation amplitude in manifold back side “1” and front side “2” for case 2. Location shown in Fig. 4.14.

1 2 f [Hz]

10 1027 183620 34 2430 2063 196340 38 3450 59 7460 2 470 3 3280 19 5490 2480 2653100 1451 816

The same characteristics were found in the computation results for all cases of the piping system and the headbox shown in Fig. 4.8. The phenomena depicted above are often found in paper machine measurements. Next, different FSI cases are compared. At most of the frequency points the velocity oscillation magnitude is of the same order in all the cases, as shown in Fig. 4.17. For the lowest frequency excitation, which is 10 Hz, all cases give high velocity oscillation amplitude, with the rigidly supported headbox model (case 4, pipe and manifold flexible) giving the lowest amplitude. In general, the rigid headbox model would give the lowest amplitude and at none of the frequencies would the basis weight variation be significant. Worth noting is that by doubling the structural thickness from 5 to 10 mm, the velocity oscillation amplitude increases at most of the frequencies. The lower wave speed case (5) gives even higher amplitudes at severe resonance peaks than the wave speed of 1000 m/s for case 2. Some researchers (Beckingham, 1993) have found that by decreasing the free air in the stock and thus increasing the wave speed, the pulsation amplitude decreases. On the other hand, others have found, that adding air to the pipe system reduces pulsation (Lewis, 1997). This discovery may be specific to a particular system, with the wave speed change also changing the natural frequencies and thus amplifying or reducing pulsation in the system. These controversial findings support the fact that no general guidelines can be given on the effect of wave speed on pulsation.

65

1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

0 20 40 60 80 100 120

f [Hz]

V/p

[m/s/

Pa]

Fig. 4.16. Relationship between velocity oscillation amplitude at headbox outlet and pressure oscillation amplitude in manifold. Finally, the velocity oscillation amplitude is plotted along the headbox outlet for case 2 for the frequencies of 10 and 50 Hz in Fig 4.18. The plots show that there may be velocity oscillation variation in the cross-direction also in the FSI case and the variation may be quite large at certain frequencies in the cross-direction. According to the variation in parameters of the paper machine model above, the headbox in this system should be supported as rigidly as possible to have lower velocity oscillation at the headbox outlet. A system with other dimensions could change the situation and the rigid headbox may not be the best choice over the whole frequency range. The modeling shows that no general guidelines can be given for the design of the approach flow system from the pulsation point of view. Each system is individual and must be designed individually.

66

0.00E+00

1.00E-01

2.00E-01

3.00E-01

4.00E-01

5.00E-01

6.00E-01

7.00E-01

0 20 40 60 80 100 120

f [Hz]

V [m

/s]

Case 2Case 5Case 3Case 4

0.00E+00

5.00E-03

1.00E-02

1.50E-02

2.00E-02

2.50E-02

3.00E-02

0 20 40 60 80 100 120

f [Hz]

V [m

/s]

Case 2Case 5Case 3Case 4

Fig 4.17. Velocity oscillation amplitude at headbox outlet, FSI cases 2, 3, 4, and 5, larger and smaller scale.

67

2.90E-012.95E-013.00E-013.05E-013.10E-013.15E-013.20E-013.25E-013.30E-013.35E-01

0 0.2 0.4 0.6 0.8 1

x [m]

V [m

/s]

10 Hz

0.00E+00

1.00E-04

2.00E-04

3.00E-04

4.00E-04

5.00E-04

6.00E-04

7.00E-04

0 0.2 0.4 0.6 0.8 1

x [m]

V [m

/s]

50 Hz

Fig. 4.18. Velocity oscillation amplitude along headbox outlet width, FSI case 2.

4.7 Real paper machine model, headbox width 10 m To obtain a better picture of what happens in a larger paper machine and to determine if the results have similar features as in the 1 m wide model, a similar system as in the previous section (Fig. 4.8) was analyzed for a headbox width of 10 m. The approach piping length was approximately 20 m long, the approach pipe diameter 0.8 m, and the pressure pulsation amplitude at the upstream end of the system 5000 Pa. The headbox was rigidly supported and the other structures had a thickness of 5 mm in the first case and 10 mm in the second case. The pressure oscillation amplitude at 40 Hz is shown for the first

68

case in Fig. 4.19. The average velocity oscillation amplitude at the headbox outlet is shown in Fig. 4.20 for case 1. The velocity oscillation amplitude in Fig. 4.20 has the same features as in the narrower headbox. At most of the excitation frequencies, the velocity oscillation amplitude is quite low, without this causing excess basis weight variation in paper. The magnitude of oscillation is very high at two frequencies, especially at 80 Hz, and that would cause too much basis weight variation and very large vibration amplitudes in the structure. The velocity oscillation amplitude at the outlet is illustrated for case 1 in Fig. 4.21 for the frequency of 10 Hz. We can see that there are variations along the cross-direction such as in the 1 m case. There is one peak at 6 m from the inlet flow edge, which is much higher than variation elsewhere in the cross-direction. At some other frequency and with other parameters the peak could be located elsewhere. Basis weight measurements for machine-direction variation are usually done by taking a strip of paper close to both edges. Basis weight measurements have shown that in some cases on one side of the machine there may be a peak at a certain frequency while on the other side there is no variation (or very little variation) at this frequency. The computations support these observations.

Fig. 4.19. Pressure oscillation amplitude in headbox of 10 m width at 40 Hz.

69

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60 80 100 120

f [Hz]

V [m

/s]

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 20 40 60 80 100 120

f [Hz]

V [m

/s]

Fig. 4.20. Velocity oscillation amplitude at headbox outlet, 10 m wide headbox, larger and smaller scale, case 1. The relationship between the pressure pulsation amplitude in the manifold and the velocity oscillation amplitude at the headbox outlet is shown in Figs. 4.22 and 4.23. For case 1 there is no simple pattern in which pressure and velocity oscillation are related to each other. In case 2, the general trend is that at lower frequencies pressure pulsation creates higher amplitude velocity oscillation than at higher frequencies. There are again some frequencies that mix up this pattern. Also, in case 2, one can have much greater pressure amplitude in the manifold to achieve the same velocity oscillation as in case 1.

70

-1.00E-01

0.00E+00

1.00E-01

2.00E-01

3.00E-01

4.00E-01

5.00E-01

6.00E-01

7.00E-01

0 2 4 6 8 10 12

x [m]

V [m

/s]

Fig. 4.21. Velocity oscillation amplitude at 10 m headbox outlet, 10 Hz.

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

0 10 20 30 40 50 60 70 80 90 100

f [Hz]

V/p

[m/s/

Pa]

Fig. 4.22. Relationship between velocity oscillation at headbox outlet and pressure oscillation in manifold, case 1.

71

0.00E+002.00E-064.00E-066.00E-068.00E-061.00E-051.20E-051.40E-051.60E-051.80E-052.00E-05

0 20 40 60 80 100 120

f [Hz]

V/p

[m/s/

Pa]

Fig. 4.23. Relationship between velocity oscillation at headbox outlet and pressure oscillation in manifold, case 2. As discussed above, the recirculation outlet has been excluded in the modeling of headboxes. The inclusion of the recirculation outlet affects the results very little. The boundary condition at the recirculation outlet is the known pressure. In the acoustical analysis the pressure is not exactly known at the outlet. A good approximation is zero pressure. When the recirculation outlet is included in case 1, the velocity oscillation amplitude at the headbox outlet differs from the results with no recirculation outlet approximately 1 % at each of the analyzed excitation frequencies, the model with recirculation giving larger values. Hence, a figure to present the differences is not needed. Therefore, the recirculation outlet can be omitted in the analysis without loss of accuracy in the analysis.

4.8 Computation requirements and mesh density The CPU time for one frequency step for an FSI model of Fig. 4.4 with 110,000 fluid elements and 40,000 structural elements computed with a 2.2 GHz workstation with 4 GB RAM is approximately 180 s. The element size in the above model is from a few to 10 cm. Furthermore, the processing time of the input file for the above case was 120 s. As we can see, even large models are solved rather quickly with an efficient computer. Nevertheless, the computation requires large memory resources and a model with 1,000,000 elements is too large to solve with the above workstation.

72

The mesh density must be sufficiently large. A mesh density dependency study was conducted for the laboratory model in Fig. 4.4. The different meshes were 1) 180,000 acoustic and 25,000 structural elements, 2) 34,000 acoustic and 8000 structural elements, and 3) 5260 acoustic and 2650 structural elements. In the rigid model meshes 1 and 2 gave identical results as seen in Fig. 4.23. In the FSI model, the mesh must be denser. The results for meshes 1 and 2 differ from each other by a few dozen percent and the difference of mesh 3 from both of these can be very large at some frequencies, as illustrated in Fig. 4.24. The scale in Fig. 4.24 is different for the rigid and FSI cases.

Rigid

0,0

0,5

1,0

1,5

2,0

0 20 40 60 80

f [Hz]

V*

mesh 1

mesh 2

mesh 3

FSI

0

5

10

15

20

0 20 40 60 80

f [Hz]

V*

mesh 1

mesh 2

mesh 3

Fig. 4.24. Dimensionless velocity oscillation amplitude (normalized with mesh 1 results) at headbox outlet for three different meshes.

73

A sufficient mesh density in FSI calculations also depends on how well the meshes match at the fluid-structure interface. If the mesh is sufficiently dense, linear elements can be employed. The results presented in this study were obtained with a sufficiently dense mesh.

4.9 Comparison of modeling results and paper machine measurement data

A comprehensive comparison of theoretical calculations and measurement data from a real paper machine cannot be made. Plenty of measurement data is available but the information for conducting pulsation or FSI analysis is inadequate. The exact length and shape of piping is needed. Also, all the details of the headbox and the manifold must be available as well as the thickness of piping and all the structures plus the supports. The exact location of pressure sensors is also unknown. In future measurements this data should be obtained to validate properly the theoretical calculations. As mentioned earlier, the wave speed changes for instance when the fabricated paper grade is changed in the approach flow system, and this change can have a drastic effect on the acoustical behavior of the system. To verify the analysis properly a set of measurements should be performed taking all of these things into account. The data available (Metso Paper, Inc.) confirms the theoretical computations. The magnitude of basis weight variation found in measurements is of the same order as the velocity oscillation at the headbox outlet at some frequencies. Several characteristics of real world measurements exhibit the same features that were obtained in the modeling and these phenomena can now be explained.

Fig. 4.25. Diagonal stripes in paper sheet.

74

The pressure sensors in a paper machine are sparsely located, usually at both extremities of the manifold and the upstream of the slice channel. The pressure amplitude may be of the same order or several times greater or smaller on different sides of the manifold. At certain frequencies the pressure oscillation is not very strong but may cause considerable basis weight variation. In some cases the situation can be vice versa, so that the high pressure amplitude is not seen in the basis weight spectrum. Very often high vibration amplitude of the headbox structure has been found to correlate with high harmonic machine-direction basis weight variation. The coupling of pulsation in the fluid and the structure vibration is obvious. In some paper measurements there have been found very visible, periodic diagonal stripes in the paper sheet, as shown schematically in Fig. 4.25. The stock approaches from the back side of the machine and therefore influences the machine-direction profile at that side earlier than at the front side (Holik, 1979). Pressure pulsation has been identified as the cause of these stripes and the distance of the stripes may correspond to the excitation from the pump or screen. Basis weight variation is usually measured on both sides of the machine. At certain frequencies, basis weight variation is found to be much stronger on one side, or there is no basis weight variation at all on the other side. The explanation is found in Figs. 4.18 and 4.21. The velocity oscillation peak could be found in some cases only on one side of the machine. All the above observations can be explained with the FSI analysis of the paper machine approach flow system and the headbox. It was shown that pressure pulsation may be in some cases one reason for variation in cross-direction basis weight.

4.10 System design aspects The reason for the high basis weight variation amplitude at some frequency that is related to the excitation frequency of a hydraulic machine does not necessarily mean a problem or malfunction in the machine but may be related to the resonance, acoustic, structural, or coupled acoustic-structural resonance of the system. To minimize basis weight variation, the hydraulic machine that creates pulsation must not be located at the anti-node position of the pressure mode shape. A change of few meters in the length in the pipeline could result in many times lower velocity oscillation at the headbox outlet. Alternatively the same effect is obtained by a change of few meters in the pump or screen location. A pipe elbow that is not correctly supported may give rise to high velocity oscillation. Computations show that high velocity oscillation

75

amplitude at the headbox outlet is frequently related to the structural vibration close to the outlet. Making the structure as rigid as possible close to the outlet seems to be favorable for low basis weight variation in paper in the machine-direction. A great challenge in designing a paper machine approach flow system with low pulsations is the effect of wave speed. The wave speed is different in different systems and varies in the same system too, depending on the mean paper basis weight to be fabricated and the paper grade, and it changes with time. The wave speed changes in the system, depending on the amount of free air, and the range can be 500-1000 m/s, or even wider. This variation changes the acoustic natural frequencies considerably. If the system has been designed for a wave speed of 1000 m/s and the stock used has the wave speed of 500 m/s, very dramatic changes can occur in pulsation. Also, pumps and screens may operate at different speeds and thus excitation frequencies are changed. Only one single frequency pulsation source has been considered in the calculations. Other sources may be added by shutting down the other sources and replacing them with a suitable boundary condition. Also phase differences may be applied. To obtain the total response, all the single frequency sources are added together. This may cause some extra effects. The effect of hydraulic machines attenuating or creating pulsation was not taken into account. Detailed modeling or measurements are needed to incorporate a hydraulic machine in the analysis.

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5 PULSATION IN PAPER COATING PROCESS A similar pulsation problem as in the approach flow system and the headbox is encountered at the end of the papermaking process, in the paper coating. Typical values for the slit of the coating machine, shown in Fig. 5.1, are the average flow velocity V = 0.2 m/s, the slit height 2h = 0.4 mm, and the slit length L = 0.1 m. The fluid density is ρ = 1500 kg/m3. The pressure pulsation amplitude pU is known at the upstream end of the slit and the fluid is discharged to the atmosphere. In this analysis the flow is considered one-dimensional. The relationship for the basis weight BW (mean), the flow velocity in the channel Vflow, and the machine speed Vpaper is obtained from the equation

paper

flow

VhV

BWρ2

= (5.1)

A typical coating basis weight for an 80 g/m2 base paper could be 6 g/m2. The paper machine wire is moving at 20 m/s and the basis weight variation in the machine-direction is 2 %, which corresponds to 2/12.0 mgBW =Δ . The velocity oscillation amplitude at the slit outlet obtained from Eq. (5.1) is 0.004 m/s.

Fig. 5.1. Paper coating principle and geometry.

77

5.1 Newtonian flow First, the flow is considered Newtonian when the constant dynamic viscosity is η = 1 kg/(ms) and the pressure pulsation frequency is 30 Hz. The pressure difference needed to drive the fluid through the slit is obtained from the pressure loss equation of Poiseuille flow

hLVfp

2

2ρ=Δ (5.2)

where VhRef h /6/6 ν== is the Moody friction factor and Eq. (5.2) becomes

2

3h

Vlp η=Δ (5.3)

The pressure difference is with the above-mentioned values Δp = 1.5⋅106 Pa. The pressure pulsation needed to create a velocity oscillation amplitude of 0.004 m/s can be calculated using the approach of harmonic fluid transients in Section 2.1.6, the method of characteristics in Section 2.1.5, the analytical solution for pulsating laminar flow Eq. (2.57), or solving the Navier-Stokes and continuity equations using CFD. All of these approaches were found to give identical results. The effect of mean flow can be incorporated in MOC and CFD approaches (also in harmonic fluid transients through quasi-steady friction in Eq. (2.40)). It was found to have no influence on the velocity oscillation at the channel outlet. To obtain the above-mentioned velocity variation of 0.004 m/s, the pressure pulsation amplitude at the slit inlet must be 6⋅104 Pa. The pulsation frequency had no effect on the results in the realistic excitation frequency range of this application (< 100 Hz). It must be noted that the ratio (pulsation amplitude)/(mean value) is double for the pressure compared to the velocity. Hence, a 4 % pressure pulsation amplitude of the mean pressure at the slit inlet causes a 2 % basis weight variation in paper. The result is also valid for other values of inlet pressure pulsation amplitude and mean value. If the viscosity is different, the percentage of pressure pulsation is still double compared to that of velocity pulsation.

78

5.2 Non-Newtonian flow If the flow is considered non-Newtonian, the material parameters corresponding to the measured data (Honkanen, 2006) for the coating paste are the consistency index k = 4.0 and the power law index n = 0.35. To obtain the velocity oscillation in this case, the problem is first tackled by solving the Navier-Stokes equations with CFD. It was also verified if the incorporating the power-law in the equations of fluid transients gives the same result as CFD computations. To obtain a velocity oscillation of 0.004 m/s with the mean velocity of approximately 0.2 m/s at the channel outlet, the pressure at the inlet was computed with the Fluent code to be ( ) Pasin103104 34 tp ω⋅+⋅= , shown in Fig. 5.2, where the pressure pulsation amplitude is 7.5 % of the mean pressure. The velocity history deviates slightly from the sinusoidal profile and is slightly inclined to the right, even though this is not very clearly seen in Fig 5.2. The mean pressure affects the velocity pulsation in the non-Newtonian fluid. With the same mean inlet pressure and varying pulsation amplitude, the mean velocity of the flow changes, which is shown in Fig. 5.3. The higher the pulsation amplitude, the lower is the mean flow velocity. The reason for this is that the shear rate is considerably different at different pressure levels. The analytical solution of Eq. (2.63) gives 0.215 m/s for steady flow, which is in accordance with the numerical results. The ratio )//()/( VVpp ΔΔ remains constant when the pressure pulsation amplitude is changed. In addition, if the pressure pulsation amplitude is kept constant and the mean pressure is increased, also the velocity oscillation amplitude increases, as shown in Fig. 5.4. The same observation, as made in the Newtonian case, is that the frequency (< 100 Hz) of pulsation does not affect the amplitude. The power-law of Eq. (2.27) was implemented both in the approach of harmonic fluid transients and MOC. It was found that in the former approach the results were very close to those obtained with CDF, the difference being less than 10 %. We can deduce that the simple model of harmonic fluid transients is sufficient for modeling pulsating non-Newtonian flow. The power-law of Eq. (2.27) was also implemented to MOC but the results were different from those of CFD, which is rather surprising. The MOC approach still needs further studying.

79

0.204

0.206

0.208

0.21

0.212

0.214

2.46 2.48 2.5 2.52 2.54 2.56 2.58 2.6 2.62 2.64

t [s]

V [m

/s]

Fig. 5.2. Velocity as function of time at slit outlet, non-Newtonian fluid.

Fig. 5.3. Mean velocity and velocity oscillation amplitude at slit outlet as function of pressure pulsation amplitude for mean pressure of 4⋅104 Pa. It is clearly a great benefit, that the simple model of fluid transients can be used to evaluate the velocity oscillation in the coating problem instead of using the more complicated CFD analysis.

80

Fig 5.4. Mean velocity and velocity oscillation amplitude at slit outlet as function of mean pressure for pressure pulsation amplitude of 6⋅103 Pa. It seems that the quasi-steady friction model is sufficient for the fluid obeying the power-law when the time scale of the problem is short. Therefore, the unsteady friction model is not needed in solving this problem. In fact, the unsteady friction model should not have any effect on the results because the coating problem does not really belong to the water hammer regime and the time scale. The results obtained above for pressure pulsation cannot be compared with experimental data, because such data is not available. The only real-world measurement data was the mean flow velocity and the basis weight variation in paper. Nevertheless, the magnitude of the pressure and the velocity is such that it could exist in reality. It is not meaningful to compare the pulsation magnitude between the Newtonian and the non-Newtonian cases because the viscosity changes all the time in the latter case. An interesting point is that the velocity oscillation amplitude depends on the mean flow velocity if the fluid is non-Newtonian. This is not the case with the Newtonian fluid. Both in the Newtonian and the non-Newtonian cases the pulsation amplitude is independent of the frequency in the realistic frequency range. In the Newtonian case, the percentage of pulsation amplitude of the mean value is double for the pressure at the slit inlet compared to the velocity at the outlet. In

81

the non-Newtonian case, the percentage of pressure pulsation is much greater than that of velocity pulsation.

82

6 CONCLUSIONS Pulsation phenomena in papermaking processes were analyzed in this study. The processes covered were: that occurring in the approach flow system and the headbox, and the paper coating process. Pulsation is created by the hydraulic machinery approximately in the frequency range of 5-100 Hz. Pulsation affects the paper quality, more specifically the basis weight variation in the machine-direction, and the machine runnability. Pulsation in papermaking processes can be modeled using different approaches. For the approach piping and the headbox, pulsation is described by the equations of fluid transients. Friction can be neglected because the pipeline is relatively short. In this case the equations of fluid transients become the wave equation of acoustics. For systems, for which rigid one-dimensional modeling is sufficient, the solution can be obtained using the approach of harmonic fluid transients. In the one-dimensional model of fluid transients also the fluid-structure interaction (FSI) may be included. Systems with three-dimensional features are modeled using a commercial FEM code of acoustics. In the FEM code, FSI can be included. The tapered slice channel of the headbox was analyzed in detail. The effect of channel shape on the velocity oscillation amplitude at the channel outlet was studied. Analytical solutions for pressure and velocity pulsation were derived from Webster’s equation of acoustics for the linear and exponential channel shapes. For other channel shapes the solution was obtained with the approximate solution of the equations of harmonic fluid transients. As mentioned above, the equations of fluid transients and acoustics become the same equation and therefore yield similar results when there are sufficient computation points in the approximate solution. It was found that by changing the slice channel shape from linear to parabolic, almost a 30 % reduction in the velocity oscillation amplitude, and therefore in basis weight variation, was obtained. The small gradient in the channel slope at the proximity of the outlet favors low velocity oscillation. It was also found, that the low contraction ratio of the channel, i.e. the taper angle, should be small to obtain low velocity pulsation amplitude. If there is free air in the liquid, the wave speed in the fluid changes with the absolute pressure change in the tapered channel. The change in wave speed does not affect the results and it is valid to assume constant wave speed. In the future, the shape could still be improved by employing a shape optimization procedure. No experimental apparatus has been made yet to verify the computations and this should be done in the future.

83

The approach piping and headboxes of a real paper machine were modeled. If the structure is assumed rigid, the 1D transmission line model, which employs the equations of harmonic fluid transients, gives very good results in terms of predicting pulsation compared to the more complicated 3D FEM model, if the machine width is small, let us say 1 m. If the width is large, of the order of 10 m, large differences are found between the 1D and 3D models. Laboratory measurements were conducted for a manifold-tapered channel system using particle image velocimetry. The measurement results, which are in good agreement with modeling results, show that the fluid-structure interaction must be taken into account to predict velocity pulsation reliably at the channel outlet. Results from the rigid model may be considerably different. The FSI analysis of paper machine approach piping and the headbox shows that high amplitude velocity oscillation, and thus basis weight variation, is related to high amplitude structural vibration at the proximity of the system outlet. Making the structure or the headbox as rigid as possible at the vicinity of the outlet could result in lower basis weight variation. Low frequency excitation seems more detrimental than higher frequency excitation for basis weight variation. If there is pressure pulsation of the same amplitude in the manifold, lower frequency pulsation causes a higher peak in the basis weight spectrum than pulsation at higher frequency. This is not valid for all the cases, because natural frequencies, both for the structure and the fluid, may affect the results at some frequencies. It was also found in the analysis that the amplitude of basis weight variation changes in the cross-direction of the machine. All of the above computational results support experimental findings. The author has performed a computational analysis for the entire paper machine approach piping and the headbox, and has shown the relationship between pressure pulsation in piping and machine-direction basis weight variation in paper. It is difficult to give guidelines whereby system design could produce low velocity oscillation amplitude at the headbox outlet. The relationship between pressure pulsation and basis weight variation is not straightforward and cannot be given as a simple formula. The changing wave speed of fluid, depending e.g. on the paper grade to be produced on the machine, and the changing excitation frequencies, i.e. the change in rotation velocity in hydraulic machines related to the change in paper grade, make it difficult to design a system from the pulsation point of view. If the aforementioned factors are fixed, the system can be designed for a particular paper machine such that high velocity pulsation amplitude can be prevented.

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The computational results of paper machine models analyzed in the study could not be comprehensively compared with any real paper machine in operation because there is no such information collected at the machines, and such information is not easy to collect when machines are in production use. In future, to carry out a comprehensive validation of results, all the information needed to conduct an FSI analysis should be collected from some paper machines. In the case of the paper coating process, including friction in the analysis is extremely important. The time scale in this problem is so slow that the equations of fluid transients are not necessarily needed to solve the problem. The solution can be obtained from the 1D equations for pulsatile flow in the slit if the fluid is Newtonian. For non-Newtonian fluid, as we have in the paper coating process, an analytical solution cannot be obtained for pulsatile flow and the problem can be solved using a CFD code. It was also found that CFD and the approach of harmonic fluid transients give very similar results for velocity oscillation at the slit outlet. It is a great benefit, that a simple model of fluid transients can be used instead of the more complicated CFD solution. In the Newtonian case the approaches of fluid transients, both in the time and frequency domain, the analytical solution of pulsatile flow, and CFD solutions give identical results. The pulsation amplitude is independent of the mean flow and the pulsation frequency in the realistic excitation frequency range. In the non-Newtonian case, the mean flow affects the pulsation amplitude and vice versa. The scientific contribution of the study is that the fluid-structure interaction analysis concerning pulsation has been done for the first time for the paper machine. All earlier studies have been restricted to rigid analysis, in which the interaction of the fluid with the structure has not been taken into account. As discussed in the theoretical section of the study, no simple guidelines can be given regarding when FSI needs to be included. This study reveals that FSI is very significant in large flexible structures such as the paper machine approach piping and the headbox. Even though a thorough comparison could not be made with measurement data, because of lack of such information, the phenomena and the characteristics found in the analysis support the experimental findings both from the pilot paper machines of 1 m width and the real production machines of 10 m width. Exhaustive validation of the computations remains a task for the future. Comparison of laboratory measurements and modeling results shows that the FSI calculations and measurements yield similar results. There have been very few validation cases, and when validation using the real paper machine measurements is carried out, we will see how well the numerical model of the Abaqus code performs. This

85

may indicate the need for development of the numerical FSI model. Furthermore, the relationship between pressure pulsation and velocity oscillation or basis weight variation in coating in the machine-direction has been shown for the first time.

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REFERENCES Abaqus Inc. 2006. Abaqus 6.6 Documentation, Providence, RI, USA. Abreu, J.M. and Betâmio de Almeida, A. 2004. Wall shear stress and flow behaviour under transient flow in a pipe. In Proceedings of the 9th International Conference on Pressure Surges, BHR Group, Chester, UK, pp. 457-476. Adamkowski, A. 2003. Analysis of Transient Flow in Pipes With Expanding or Contracting Sections. Journal of Fluids Engineering, 125, pp. 716-722. Arenas, J.P. and Crocker, M.J. 2001. Approximate Results of Acoustic Impedance for a Cosine-Shaped Horn. Journal of Sound and Vibration, 239(2), pp. 369-378. Axworthy, D.H., Ghidaoui, M.S. and McInnis, D.A. 2000. Extended Thermodynamics Derivation of Energy Dissipation in Unsteady Pipe Flow. Journal of Hydraulic Engineering, 126(4), pp. 276-287. Beckingham, G.E. 1993. Methods for Reducing Pulsations in the Approach Flow System. Progress ’93, Lodz, Poland, Sept. 27-30, 1993. Lodz, Poland, Association of Polish Papermakers, pp. 138-147. Belardinelli, E. and Cavalcanti, S., 1992. Theoretical Analysis of Pressure Pulse Propagation in Arterial Vessels. Journal of Biomechanics, 25(11), pp. 1337-1349. Blevins, R.D. 2001. Flow-induced vibration. Krieger Publishing Company, Florida. Brunone, B., Golia and U.M., Greco, M. 1991. Some remarks on momentum equation for fast transients. Proceedings of the International Meeting on Hydraulic Transient and Column Separation, IAHR, Valencia, Spain, pp. 201-209. Brunone, B., Karney, B.W., Mecarelli, M. and Ferrante, M. 2000. Velocity Profiles and Unsteady Pipe Friction in Transient Flow, Journal of Water Resource Planning Management, 126(4), pp. 236–244.

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APPENDIX A 1D FSI equations for pipe section FSI in a straight pipe is governed by 14 1D partial differential equations. Extended equations of fluid transients The extended equations of fluid transients

01=

∂∂

+∂∂

zp

tV

fρ (A.1)

( )z

utp

EeR

KzV z

∂∂

=∂∂

⎥⎦⎤

⎢⎣⎡ −++

∂∂ &

νν 2211 2 (A.2)

govern the unknowns of pressure, p, fluid velocity, V, and axial pipe velocity,

zu& . fρ is the fluid density, z the z-coordinate, K the bulk modulus of fluid, ν the Poisson constant, R the pipe radius, E the Young’s modulus and e the pipe wall thickness. The classical equations of fluid transients are obtained in the case 0=ν .

Fig. A.1. Geometry and coordinates of straight pipe section.

92

Axial motion of pipe The extended beam equations

01=

∂∂

−∂∂

ztu z

s

z σρ

& (A.3)

tp

EeR

tEzu zz

∂∂

−=∂∂

−∂∂ νσ1&

(A.4)

govern the axial motion of a pipe where sρ is the density of structure and zσ the axial pipe stress. Lateral motion The Timoshenko beam equations describe the flexural vibration in the y-z plane

01=

∂

∂

++

∂

∂

zQ

AAtu y

ffss

y

ρρ

& (A.5)

xy

s

y

tQ

GAtu

θκ

&&

−=∂

∂+

∂

∂21 (A.6)

yss

x

ss

x QIz

MIt ρρ

θ 11=

∂∂

+∂∂ &

(A.7)

01=

∂∂

+∂∂

tM

EItx

s

xθ& (A.8)

with Qy the lateral shear force, yu& the lateral pipe velocity, xM the bending

moment, and xθ& the rotational pipe velocity. Af and As are the cross-sectional areas of fluid and pipe, Is the second moment of inertia, G the shear modulus of pipe wall material, and ( ) ( )ννκ 34/122 ++= the shear coefficient. The equations governing lateral vibration in the x-z plane are

93

01=

∂∂

++

∂∂

zQ

AAtu x

ffss

xρρ

& (A.9)

yx

s

xt

QGAt

uθ

κ&&

−=∂∂

+∂∂

21 (A.10)

xss

y

ss

y QIz

MIt ρρ

θ 11=

∂

∂+

∂

∂ & (A.11)

01=

∂

∂+

∂

∂

tM

EIty

s

yθ& (A.12)

Torsional motion The torsional equations

01=

∂∂

−∂∂

zM

Jtz

ss

zρ

θ& (A.13)

01=

∂∂

−∂∂

zM

GJzz

s

zθ& (A.14)

govern the unknowns of torsional moment, zM , and torsional angular velocity, zθ& , and Js is the polar second moment of area.

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