Methods for Reducing the Complexity of Geometrical

Methods for Reducing the Complexity of Geometrical
Methods for Reducing the
Complexity of Geometrical
Structures Based on CFD
Time Efficient Simulations Based on Volume Forces
Coupled with Single and Two-phase Flow
Kamal Rezk
Faculty of Health, Science and Technology
Environmental and Energy Systems
DISSERTATION | Karlstad University Studies | 2014:32
Methods for Reducing the
Complexity of Geometrical
Structures Based on CFD
Time Efficient Simulations Based on Volume Forces Coupled
with Single and Two-phase Flow
Kamal Rezk
DISSERTATION | Karlstad University Studies | 2014:32
Methods for Reducing the Complexity of Geometrical Structures Based on CFD
Programming - Time Efficient Simulations Based on Volume Forces Coupled with
Single and Two-phase Flow
Kamal Rezk
Karlstad University Studies | 2014:32
ISSN 1403-8099
ISBN 978-91-7063-565-6
The author
Karlstad University
Faculty of Health, Science and Technology
Department of Engineering and Chemical Sciences
SE-651 88 Karlstad, Sweden
+46 54 700 10 00
Print: Universitetstryckeriet, Karlstad 2014
Throughout recent years, computer based programs have been applied to solve
and analyse industrial problems encountered in global fields such as automobile
design for reduction of CO2-gas and designing wind parks aimed at increasing
power output. One of these developed programs is Computational Fluid
Dynamics (CFD) which numerically solves complex flow behaviour based on
computer power.
As there is an ongoing expansion of CFD usage in industry, certain issues need
to be addressed as they are becoming more frequently encountered. The general
demand for simulations of larger control volumes and more advanced flow
processes result in an extensive requirement of computer resources. Moreover,
the implementation of commercial CFD codes in small-scaled industrial
companies seems to generally be utilised as a black box based on the knowledge
of fluid mechanic theory. Increased partnerships between industry and the
academic world involving various CFD based design processes generally yield
to a verbal communication interface, which is a crucial step in the process given
the level of dependency between both sides.
Based on these notions, a method for establishing time efficient CFD-models
with implementation of volume forces as sink terms in the momentum
equation is presented. The internal structure, or parts of the structure, in the
simulation domain is removed which reduces the geometrical complexity and
along with it, computational demand. These models are the basis of assessing
the benefits of utilizing a numerical based design process in industry in which
the CFD code is used as a communication tool for knowledge sharing with
counterparts in different fields.
Summary in Swedish
Genom åren har datorbaserade program applicerats för att lösa och analysera
industriella problem som exempelvis design av bilar för att minska CO2-utsläpp
samt utformning av vindkraftsparker som syftar till att öka producerad effekt
osv. Ett av dessa program heter Computational Fluid Dynamics (CFD) som
numeriskt löser komplexa flödesprocesser baserat på datorkraft.
I och med den konstanta utbredningen av CFD i industrin, måste flera punkter
adresseras av den orsaken att dem förekommer ofta. Generella krav för
simulering av stora kontrollvolymer med avancerade strömningsförhållanden
kräver omfattande datorresurser. En till viktig punkt är att användningen av
CFD i småskaliga industrier har en tendens att användas som en ”black box”
modell utifrån vetenskapen av strömningsmekanik. Ett ökat samarbete mellan
företag och universitet i CFD-baserade designprojekt leder till problematiken
runt kommunikationen mellan olika parter med olika vetenskapliga profiler.
Utifrån dessa iakttagelser presenteras en metod för att fastställa tidseffektiva
CFD-modeller genom implementering av volymkrafter som källtermer i
rörelsemängdsekvationen. Den inre strukturen, eller delar av strukturen i
simuleringsdomänen tas bort som minskar den geometriska komplexiteteten
och i sin tur, behov av datorprestanda. Dessa modeller är grunden för att
bedöma fördelarna med att använda en numeriskbaserad designprocess inom
industrin där CFD koden används som ett kommunikationsverktyg för
kunskapsdelning mellan professioner i olika områden.
Summary of Papers
In paper I, a heuristically determined design process of the geometry near the
front trap door of an internal duct system of a heat pump tumble dryer was
achieved by implementing the CFD code COMSOL MultiPhysics as a
communication tool. The design process was established by two counterparts in
the project, in which CFD calculations and geometry modifications were
conducted separately. Two design criteria presenting the pressure drop in the
duct and the outflow uniformity were used to assess geometry modifications
conducted by a mechanical engineer. The geometry modifications were based
on visual results of the flow patterns. The modifications confirmed an
improvement in the geometry as the pressure drop was reduced by 23%, and
the uniformity was increased by 3%.
In paper II, single-phase simulations were conducted on a tube-fin heat
exchanger in order to simulate airflow. The heat exchanger is regarded as a
porous medium in which the arrangement of fins and tube bundles are replaced
with volume forces such as sink terms in the momentum equation. Hence, the
computational time was reduced significantly for the structureless model. The
focus was on achieving a correct volume flow rate and pressure drop relation.
Moreover, experimental results of the flow rate and pressure drop relation
showed good agreement with the volume force model.
In paper III, a numerical model of a vacuum dewatering process was
established by using a two-phase flow model. In this study, a 2-dimensional
model of the paper sheet was analysed numerically using a Level-Set method,
coupled with the continuity and momentum equations. The purpose of
analysing vacuum dewatering numerically is to gain a physical understanding of
the process and eventually to obtain better methods of dimensioning industrial
equipment. The aim was to calculate the dry content and the flow rates of water
and air as a function of dwell time and vacuum levels. Simulations of vacuum
dewatering of paper with a basis weight of 50 g/m2 showed somewhat
satisfactory results compared to experimental data, in terms of relating the dry
content of the paper to the dwell time.
In paper IV, the numerical model presented in paper III was further developed
by considering 3-dimensional flow resistance in a 2-dimensional two-phase flow
model. A MATLAB algorithm linked to the CFD program COMSOL was
created in order to execute numerous steps in the simulation process in a selfgoing manner. The algorithm generated fibers with randomly distributed
coordinate positions and orientations in a sample domain representing a paper
sheet. Single-phase simulations of air and water were conducted for various Renumbers. A numerical data set was established and used to estimate flow
resistance as volume forces for the sample domains representing the paper
sheet and the forming wire. These volume forces were implemented in a 2dimensional two-phase flow model, in which the dry content and dewatering
rate were related to the dwell time. The porosity variation in the paper sheet
and the influence of the fibers packing on the forming wire were tested
numerically. The numerical simulations showed that the packing evolution of
the fibers on the forming wire presented the most pleasing results compared to
the experimental data.
In paper V, a MATLAB algorithm linked to the CFD program COMSOL was
created, in order to produce sample domains with various arrangements of
cylindrical objects representing fibrous materials. The flow resistance in 3dimensional fibrous structures was investigated in a wide range of Re-numbers,
in which inertia was included. Resistance coefficients were established based on
steady state simulations of single-phase processes of water numerically. The
numerical data set were used to estimate Forchheimer coefficients, which are
used to correlate a dimensionless friction factor to a modified Reynolds
expression for porous media. The friction factor and dimensionless
permeability were calculated for fibrous arrangements in a 3-dimensional space.
The numerical simulations agreed well to classical empirical formulations,
regardless of their fiber orientation and porosity. The estimation of the friction
factor is, however, sensitive depending on the range of Re-numbers over which
the resistance coefficients are estimated over. The viscous resistance was
estimated with better accuracy, as the dimensionless permeability was in good
agreement with the current simulation data from other literature.
I received my Master of Science degree in Environmental and Energy Systems
at Karlstad University in 2007. I began my PhD studies at Karlstad University
in the autumn of 2008.
This thesis originates from the cooperation between Karlstad University and
the household appliance company Asko Appliance AB, which also financed the
project that resulted in a licentiate degree. The aim of the cooperation was to
establish a good relation between the industry and the university. By doing so,
several projects have been created with the purpose of sharing knowledge and
creating a foundation for applied academic research in the department.
I acquired my licentiate in February 2011, with the thesis labelled as “CFD as a
tool for analysis of complex geometry”. The thesis addressed the beneficial
aspects of implementing CFD analysis in a design cycle consisting of
counterparts with different specialist fields. The focus was to point out the
CFD code as a communication tool and a foundation for knowledge sharing.
Also, the thesis introduced the concept of implementing volume forces as sink
terms in the momentum transport equation in order to reduce computational
demand. The concept was introduced by predicting the flow rate and pressure
drop relation of a fin-tube heat exchanger.
My PhD research continued at Karlstad University through exploration of the
volume force concept by characterising flow resistance in a vacuum dewatering
process of paper sheets. The method was further explored by comparing the
flow resistance of the volume force terms to classically empirical formulations,
other numerical simulations as well as experimental data.
Parts of the text in my licentiate thesis are reused in this work which consists of
background facts, theory and historical inputs in the subject of fluid mechanics.
Karlstad, May 2014
I would like to start by thanking Asko Appliance AB for financing the projects
concluding paper I and II. Thanks to Peder Bengtsson and Anders Sahlén for
having faith in me.
Thanks to Mechanical Design Engineer Johan Olsson from Asko Appliance
AB for conducting geometry modifications in paper I.
I would like express my gratitude Assoc Prof Ola Holby for supervising me
during the first half of this project which concluded paper I and II.
I want to thank Lars Pettersson, Laboratory Assistant, for assembling
laboratory setups for paper II.
I want to thank my supervisor Jonas Berghel who has been a big support for
from the start of my studies. I am very grateful and thank you for your valuable
I want to thank my supervisor Lars Nilsson who has been with me since my
licentiate. I have learnt a great deal from you and your enthusiasm has been a
great inspiration for me.
I want to express my gratitude to Jan Forsberg who has been with me from the
start of my studies. Thank you for the time you have put in. Our conversations
have been a great inspiration for me personally and your encouragement has
been immeasurable.
I wish to thank my colleagues at the Department of Energy, Environmental and
Building Technology at Karlstad University for creating an inspiring
environment for learning.
Finally, I would like to thank my family: my loving girlfriend for being a big
support even though during a short time span of the project; my big brother for
showing enthusiasm and challenging me with smart questions; my little sister
for being my biggest fan and my biggest support in terms of enduring my
frustration periods.
And to my two greatest inspirations: my parents. My father for being proud of
me and not letting a day go by without asking me about my studies. My mother
who is the strongest person I know and the greatest source for me to find
encouragement and strength.
‫إلى والدي‬
‫هما مين و إحنا مين‬
‫‪by Ahmed Fouad Negm‬‬
‫هما مين و إ حنا مين ‪ ..‬؟‬
‫هما األمرا و السالطين ‪..‬‬
‫هما المال و الحكم معاهم ‪..‬‬
‫و إحنا الفقرا المحرومين‬
‫حزر فزر ش ّغـل مخك‬
‫شوف مين فينا بيحكم مين ؟‬
‫إحنا مين و هما مين ‪..‬‬
‫إحنا الفعال البنايين‬
‫إحنا السنة و إحنا الفرض‬
‫إحنا الناس بالطول و العرض‬
‫من عافيتنا تقوم االرض ‪..‬‬
‫و عرقنا يخضر بساتين‬
‫حزر فزر شغل مخك‬
‫شوف مين فينا بيخدم مين ‪..‬؟‬
‫هما مين و إحنا مين ؟‬
‫هما االمرا و السالطين‬
‫هما الفيال و العربية‬
‫و النساوين المتنقية‬
‫حيوانات استهالكية‬
‫شغلتهم حشو المصارين‬
‫حزر فزر شغل مخك‬
‫شوف مين فينا بياكل مين ‪..‬؟‬
‫إحنا مين و هما مين‬
‫إحنا قرنفل على ياسمين‬
‫إحنا الحرب ‪ :‬حطبها و نارها‬
‫و إحنا الجيش اللي يحررها‬
‫و إحنا الشهدا بكل مدارها‬
‫منكسرين أو منتصرين‬
‫حزر فزر شغل مخك‬
‫شوف مين فينا بيقتل مين ‪..‬؟‬
‫هما مين و إحنا مين‬
‫هما االمرا و السالطين‬
‫هما مناظر بالمزيكا‬
‫و الزفة و شغل البوليتيكا‬
‫و دماغهم طبعا استيكا‬
‫بس البركة بالميازين‬
‫حزر ‪ ..‬فزر شغل مخك‬
‫شوف مين فينا بيخدع مين ‪..‬؟‬
‫هما مين و إحنا مين‬
‫هما االمرا و السالطين‬
‫هما بيلبسوا أخر موضة‬
‫و إحنا بنسكن سبعة في أوضة‬
‫هما بياكلوا حمام و فراخ‬
‫و إحنا الفول دوخنا و داخ‬
‫هما بيمشوا بطيارات‬
‫و إحنا نموت باالوتوبيسات‬
‫هما حياتهم بامب جميلة‬
‫هما فصيلة و إحنا فصيلة‬
‫حزر فزر ‪ ..‬شغل مخك‬
‫شوف مين فينا حيغلب مين‬
‫‪A poem by late Egyptian vernacular poet Ahmed Fouad Negm. The translation of the title is “Who‬‬
‫‪are they and who are we”.‬‬
Force field
Source term (COMSOL)
Surface tension
Unit tensor
Curve element
Normal vector
Velocity field
Drag coefficient
Dimensionless friction factor
Diffusion coefficient (COMSOL)
Surface tension
Piecewise interpolant
Gravitational acceleration
Neumann condition
Mesh cell length
Weight coefficient (COMSOL)
Characteristic length of pore size
Characteristic length scale of porous media
Mass flux
Number of points
Heat flux
radii of tube
Dirichlet condition
Specific surface area
Intrinsic velocity
Velocity magnitude
Superficial velocity
Local velocity coefficient
Transport quantity (COMSOL)
Propagation velocity of a curve
Volume flow rate
Space coordinate
Space coordinate at the interface boundary
Cartesian coordinates
Surface function
Greek symbols and mathematical operators
( )
Convective transport term (COMSOL)
Viscous resistance coefficient
Volume fraction fluid phase
Convective velocity vector
Inertial resistance coefficient
Source/sink term (COMSOL)
Re-initialisation parameter
Difference term
Dirac delta function
Interface thickness
Dynamic viscosity (COMSOL)
Brinkman screening length
Curvature of interface
Dynamic molecular viscosity
Slip length
Surface tension coefficient
Level-Set function
Basis function
Partial derivative
Gradient operator
Volume-averaged property
Tensor indices
Cartesian coordinates
Constant properties (COMSOL)
Surface tension
Computer aided design
Computational fluid dynamics
Degrees of freedom
Finite difference method
Finite element method
Flow representative volume
Finite volume method
Galerkin finite element method
International Energy Agency
Partial differential equation
Representative elementary volume
Volume-averaging theory
Volume force
List of Publications
This dissertation is based on the following papers:
Paper I:
Rezk K. and Forsberg J. (2010). Geometry development of the
internal duct system of a heat pump tumble dryer based on fluid
mechanic parameters from a CFD software. Applied Energy.
88(5): 1596-1605
Paper II:
Rezk K. and Forsberg J. A fast running numerical model based on
the implementation of volume forces for prediction of pressure
drop in a fin tube heat exchanger. Accepted for publication in
Applied Mathematical Modelling, April 2014
Paper III:
Rezk K. Nilsson L. Forsberg J. Berghel J. (2013). Modelling of
water removal during a paper vacuum dewatering process using a
Level-Set method. Chemical Engineering Science. 101(0): 543-553
Paper IV:
Rezk K. Nilsson L. Forsberg J. Berghel J. Simulation of water
removal in paper based on a 2D Level-Set model coupled with
volume forces representing fluid resistance in 3D fiber
distribution. To be resubmitted with a minor revision to Drying
Paper V:
Rezk K. Nilsson L. Forsberg J. Berghel J. Characterizing flow
resistance in 3-dimensional disordered fibrous structures based on
Forchheimer coefficients for moderate Reynolds numbers.
Manuscript ready for submission.
The following related publication is not included in this thesis:
Rezk K. Nilsson L. Forsberg J. Berghel J. Using a Level-Set model to estimate
dwell time in a vacuum dewatering process for paper. Oral presentation at the
COMSOL conference in Milan 2012.
The Author’s Contribution
Paper I:
Planning was done on equal parts. All writing was performed by
me. Conducted simulations were performed by me with Jan
Forsberg as a discussion partner. Ola Holby and Jonas Berghel
contributed with valuable comments.
Paper II:
Planning was done on equal parts. All writing was performed by
me, except the experimental procedure, which was written by Jan
Forsberg. Conducted simulations were performed by me with Jan
Forsberg as a discussion partner. Measurements were performed
along with Jan Forsberg. The sensitivity study of the experimental
process was conducted by Jan Forsberg. Ola Holby, Jonas Berghel
and Lars Nilsson contributed with valuable comments.
Paper III:
Planning was done on equal parts. Most of the writing was
performed by me with the help of Lars Nilsson in the introduction
and paper sheet characterisation. The conducted simulations were
performed by me with Jan Forsberg as a discussion partner.
Measurements and compilation of experimental data from the
literature were performed by Lars Nilsson. Lars Nilsson, Jan
Forsberg and Jonas Berghel contributed with valuable comments.
Paper IV:
Planning was done on equal parts. All writing was performed by
me. Conducted simulations were performed by me with Jan
Forsberg as a discussion partner. Lars Nilsson, Jan Forsberg and
Jonas Berghel contributed with valuable comments.
Paper V:
Planning was done on equal parts. All writing was performed by
me. Conducted simulations were performed by me with Jan
Forsberg as a discussion partner. Lars Nilsson, Jan Forsberg and
Jonas Berghel contributed with valuable comments.
Table of Contents
ABSTRACT .................................................................................................................1
SUMMARY IN SWEDISH .....................................................................................2
SUMMARY PAPERS ................................................................................................3
PREFACE ....................................................................................................................5
ACKNOWLEDGEMENTS ....................................................................................6
NOMENCLATURE .................................................................................................9
LIST OF PUBLICATIONS ................................................................................. 12
THE AUTHOR’S CONTRIBUTIONS .......................................................... 13
TABLE OF CONTENTS .................................................................................... 14
1. INTRODUCTION .......................................................................................... 16
1.1. Objectives ...................................................................................................... 18
2. BACKGROUND ............................................................................................... 20
2.1. Commercial CFD codes .............................................................................. 20
2.2. Communication interface ................................................................................ 22
2.3. Volume-averaging theory ............................................................................... 24
3.1. Heat pump tumble dryer ................................................................................ 26
3.2. Papermaking ...................................................................................................... 27
3.2.1. Paper machine.......................................................................................... 28
3.2.2. Vacuum dewatering ................................................................................. 29
3.2.3. Forming wire ........................................................................................... 30
4. THEORY OF FLUID MECHANICS........................................................ 32
4.1. Pressure in fluid flow .................................................................................. 33
4.2. Navier-Stokes equations .............................................................................. 37
4.3. Accounting for turbulence ........................................................................ 38
4.4. Two-phase flow ......................................................................................... 39
4.5. Flow in porous media .................................................................................. 41
4.5.1. Fluids and porous media as continua ....................................................... 41
4.5.2. Flow representative volume....................................................................... 42
4.5.3. Reynolds number .................................................................................. 43
4.5.4. Viscous and inertial contributions in porous media .................................. 46
5. METHOD .......................................................................................................... 49
5.1. Modelling in COMSOL ............................................................................... 49
5.1.1. Finite Element Method .......................................................................... 49
5.1.2. Numerical instabilities ........................................................................... 51
5.1.3. Boundary conditions............................................................................... 53
5.1.4. Level-Set method ................................................................................... 54
5.2. Volume forces coupled to single and two-phase flow ............................ 57
5.2.1. Pore-scale simulations ............................................................................... 57
5.2.2. Modelling surface tension .......................................................................... 59
5.2.3. Closure model .......................................................................................... 60
5.2.4. Sensitivity studies ..................................................................................... 62
5.2.5. Volume forces in structureless models ........................................................ 65
5.3. Characterisation of flow resistance in various fibrous structures .......... 68
5.3.1. Algorithm for production runs in COMSOL ........................................... 68
5.3.2. Assessing flow resistance based on Forchheimer coefficients ......................... 71
RESULTS AND DISCUSSION ................................................................... 73
6.1. Fitting procedure of resistance coefficients: Paper II, III, IV, V .......... 73
6.1.1. Concluding remarks ................................................................................. 75
6.2. Implementation of volume forces as sink terms: Paper II, III, IV ....... 75
6.2.1. Concluding remarks ................................................................................. 81
6.3. Sensitivity study of pore-scale simulations: Paper II, III, IV, V ............ 82
6.3.1. Concluding remarks ................................................................................. 84
6.4. Experimental validation of simulation models: Paper II, III, IV .......... 85
6.4.1. Concluding remarks ................................................................................. 86
6.5. Assessment of flow resistance coefficients: Paper V .............................. 87
6.5.1. Concluding remarks ................................................................................. 90
6.6. Assessment of numerical based design process: Paper I ........................ 91
6.6.1. Concluding remarks ................................................................................. 93
CONCLUSSIONS ............................................................................................ 94
FUTURE RESEARCH ................................................................................... 96
REFERENCES ................................................................................................. 97
In recent years, computer based programs, for instance various commercially
available CFD tools, have been applied to solve and analyse industrial
problems. CFD is the abbreviation for Computational Fluid Dynamics, which
is one of the branches of fluid dynamics. Increased developments of highperformance computer hardware, as well as the introduction of user-friendly
interfaces, have paved the way for the extensive application of CFD in various
fields. Such fields include the aerodynamics of aircrafts and vehicles (reducing
the drag coefficient of a car for lower fuel consumption), chemical process
engineering (improving the mixing capability of a static mixer, which is
applicable in wastewater treatment or refinery in the oil and gas market),
biomedical engineering and drying technology. Drying technology is a typical
multidisciplinary topic that covers fields such as the pulp and paper industry as
well as the agricultural industry to name a few.
By way of a recent progression of computer efficiency, several aspects of CFD
usage have been compiled into an advantageous approach over experimentalbased approaches in fluid systems’ design that results in:
 the possibility to execute detailed analyses of a system;
 the capability to simulate large control volumes, which present
difficulties in physical laboratories;
 a possible tool for communication of results, based on detailed
visualisation images;
 a time and cost efficient tool in various fluid design processes.
Even though a numerical design process is not capable of replacing pilot-scale
testing completely, it reduces the expense due to a reduced need of
manufacturing prototypes for measurements, and fewer rentals of large
experimental facilities and measurement equipment. Large industrial companies
generally encompass an experienced CFD department with vast full-scale
experimental laboratories. In addition, they are in possession of computer
cluster networks. Numerous small-sized companies, however, lack experience
in the subject, computer resources and extensive experimental facilities.
User-interfaces in CFD programs have developed rapidly. Yet, the history of
research concerning the theory of fluid dynamics is extensive. Consequently,
CFD software in small-sized industrial companies is utilised as a black box,
based on the knowledge of fluid mechanic theory. With the availability of userfriendly interfaces, CFD calculation can be conducted on a practical basis in
various industries. Yet again, a crucial part in the simulation process is to
evaluate data by using visual analysis of flow patterns, analysis on the sensitivity
of the mesh grid and investigation of quantitative parameters such as pressure
loss, velocity, turbulence intensity, etc. There is, however, a need for
collaboration involving the industry and the academic world in order to carry
out applied research towards the objectives of the industry. A direct partnership
involving researchers and industry generally yields to a verbal communication
interface, which is a crucial step in the process given the level of mutual
In this study, the aim is focused on the methods of reducing the geometrical
complexity of systems involving the transportation of mechanical energy. In an
engineering design cycle, there is a demand for quick analysis in order to
evaluate the modified design and performance of the product or process. Even
though high computer power is accessible these days, there is a need for time
efficient simulations. The most common approach, which basically every CFD
engineer is utilising, is to divide the computational domain into several
subdomains in order to reduce the total amount of Degrees of Freedom
(DOF). A crucial aspect in this process is determining where the boundaries of
these subdomains are most representable. A continually evolving method in
reducing DOF in simulation models is volume-averaging constitutional
variables such as velocity, pressure and temperature, to name a few. With this
method, volume forces that represent flow resistance can replace complex
geometrical configurations, which will reduce the total amount of DOF while
solving the flow problem. This enables time efficient simulations of systems so
that various geometrical configurations and process parameters could be
analysed numerically.
This study originates from the fact that there is a need to utilise the
communication process between the different professions in a numerical based
design process, and establish fast running numerical models based on volume
forces that represent flow resistance in geometrically complex structures. The
fields of application in this thesis are:
Improving the aerodynamic capabilities of the internal duct system in a
heat pump tumble dryer.
Analysis of process parameters in various running conditions in a
vacuum dewatering process in a paper machine.
The volume-averaging method is not limited to these application fields,
however, as the prospect of reducing geometrical complexity is achievable in
many industrial applications. Moreover, the demand for fast running numerical
models is crucial in various design processes in both small and large scale
1.1 Objectives
The objectives of this thesis is to investigate possible ways of reducing the
complexity of geometrical structures in terms of minimising the computational
time. The essence of the investigation is to present possible methods based on
the capacity of standard personal computer hardware. For this, two methods
are utilised:
Analysis of the early implementation of CFD analysis in a numerical lab
design process, in which systems’ complexity is reduced by appropriately
subdividing the system into trivial components.
Implementation of volume forces as sink terms representing flow
resistance in the momentum equation for a single- and two-phase flow.
In (i), CFD software is utilised as a communication tool during a numerical
based design process, in cooperation with a household appliance company. The
design process is conducted through a communication interface between the
researcher and a mechanical engineer, which is a representative of the company.
The focus in the first point, (i), is:
1. Assess the contribution of early implementation of CFD
simulations in a numerical based lab design cycle process.
The second point, (ii), investigates the validity of implementing volume forces
in systems thereby regarding them as porous media. The first application area is
simulating pressure drop and flow rate relations in a staggered pin-fin heat
exchanger as an integral component of a heat pump tumble dryer. The second
application area is simulating a vacuum dewatering process based on
representing the flow resistance of the fibrous structure of paper and the woven
forming wire with volume forces. The focus in the second point, (ii), is:
1. Validating system parameters based on volume force
implementation in structureless models. Moreover, analysing
the sensitivity of these models based on mesh configuration
and numerical sample points used to estimate flow resistance.
2. Validating volume force implementation and flow resistance
characterisation to experimental and literature data.
3. Assessing the contribution of volume force implementation in
analysed systems in terms of evaluating computational time.
This chapter will provide a brief survey of commercial CFD usage, the
importance of CFD based design processes and a view on the utilisation of
CFD implementation in companies with a lack of experience in the field of
fluid mechanics. Moreover, a review of the volume-averaging method is
2.1 Commercial CFD codes
A CFD program uses numerical methods to implement a discretisation on
Partial Differential Equations (PDE) and to solve them. The purpose of using
CFD is to solve complex fluid flow relations, which is not feasible with
standard calculus. With the help of a CFD program, it is possible to simulate
and visualise; e.g. the airflow around the wing of an aircraft to determine the
drag and lift force on the wing, or the flow in a fuel cell stack to determine
whether the flow and the design of the channels are proper for a certain
Most commercial CFD codes involve three main elements:
• Geometry design
• Mesh grid
• Physical
• 2D and 3D surface
• Vector and
streamline plots
• Display of domain
Fig.(1). Representative elements for a general CFD code.
The pre-processor constitutes a computational domain, which can be designed
by the user in the CFD code or imported from an external Computer Aided
Design (CAD) program. Mesh grid generation is an essential part of the
simulation process. It represents a grid that divides the computational domain
into subdomains, in which the PDEs are discretised and solved numerically at
the nodes of the grid intersections. In general, large numbers of cells wield
better solution accuracy. Numerous commercial codes comprise their own
mesh grid generation features and major CFD codes are compatible with
external CAD and mesh generator programs. Hence, more advanced mesh grid
structures can be imported to the CFD codes. In recent CFD codes, predefined
PDEs, along with material property libraries are accessible to the user. Such
PDEs include various physical processes such as energy, mass and momentum
The Finite Difference Method (FDM) is one of the methods used to discretise
the equations of flow for a computational solution. The method approximates
the solution to algebraic equations using the Taylor-expansion series through a
forward, backward and/or central difference scheme. The Finite Volume
Method (FVM) discretises a small volume surrounding a node point on a mesh.
This method generates conservative PDEs, since the flux that enters a given
control volume is identical to the one that leaves the adjacent volume, due to
the divergence theorem. In the Finite Element Method (FEM), the PDEs over
the elements are approximated by local functions such as polynomials, resulting
in a linear system of algebraic equations. The boundary and initial value
conditions are formulated in either a weak or integral form. The subdivided
elements are unique, as the grid could be shaped as tetrahedrons or
hexahedrons for a 3-dimensional domain. This is a result of the fact that the
FEM becomes more flexible as more complicated geometries can be treated.
One of the most crucial steps when utilising a commercial CFD code is
evaluating the data. Due to increased implementation of commercial codes in
the industry, the graphic capabilities of such programs have vast and versatile
data visualisation toolboxes. Output data can be presented as 2-dimenisonal
and 3-dimensional with colour, vector, contour lines and streamline plots.
Typical output data includes velocity patterns, heat flux, temperature and
pressure distribution.
2.2 Communication interface
The communication process of various fields of knowledge is an issue involving
the integration of CFD calculation and structural geometry design. The
complexity involving CFD usage leads researchers such as Koikekoi [1], to study
strategies of practical engineering software applications in order to amplify the
implementation of software for the next generation. A concept of an interface
instrument for different professions was proposed by Stenzel and Pourroy [2].
The implementation focuses on product knowledge as a tool to initiate
communication between professions and to support their learning about the
product. Petridis and Knight [3] acknowledged the difficulties of implementing a
CFD software package into a so-called Intelligent Knowledge-Based System.
They suggested a blackboard model to facilitate the use of CFD data by various
engineering groups.
A numerical based design process can be utilised in different ways. Numerical
based design of various components is one of many study fields that has
evolved, along with the development of CFD codes and improved computer
power. There are several ways to carry out geometrical improvements through
CFD calculations. El-Sayed et al. [4] integrated an optimisation code into a CFD
code to create a shape optimisation tool to study the design of an aerofoil and
an S-shaped duct. The aerodynamic performance of a wing has been optimised
by combining the CFD software Fluent and the iSIGHT design platform [5].
Another method for designing a geometrical structure is to visually analyse the
airflow patterns in the simulation model. In an article Yakinthos et al. [6], a 2dimensional approach was used to solve the Launder-Sharma low-Reynolds k-ε
model in order to capture recirculation regions close to the walls of heat
exchangers installed in a recuperative aero engine. Several configurations were
investigated to achieve the lowest value of the pressure drop. Lakshmiraju and
Cui [7] implemented a k-ε model in the CFD software Fluent. Detailed data was
established on the flow field in order to analyse the mechanisms of pressure
loss in a power plant stack model.
A typical numerical based design process involving several professions is
described in Fig (2). It is viewed as a deterministic algorithm, considering the
fact that each procedure is operated under the same considerations. Geometry
modifications are heuristically determined based on visual images of flow
patterns and design criteria for the model.
Simulation process
Geometry modification
by the company’s
mechanical engineer
Evaluation of flow parameters
Design criteria
Fig. (2). Schematic description of the design process.
Due to its complex geometric structure, considerations of how fluid mechanic
parameters are visually presented are critical in order to amplify the
communication. It is essential to present a high spatial resolution in the domain
that provides the mechanical engineer with detailed information on where
unfavourable flow behaviour is generated. The evaluation of the flow
parameters is the criterion that determines the modification to the geometry.
The modifications are confirmed at the end of the design loop by comparing
the design criteria with the previous model. The design criteria are usually
coupled to integrating constitutional variables such as temperature, pressure
and velocity. The criteria can vary depending on the system at hand, e.g. heat
flow depends on the geometrical configurations of a heat exchanger, and the
dewatering rate depends on the arrangement of the fibers constituting the paper
The design criteria and visual data are the foundation of the communication
process. It is preferable to communicate the CFD results verbally. Otherwise,
visual data of flow patterns can be misinterpreted. The communication is a
continuous process and it is looped until further modifications are no longer
2.3 Volume-averaging theory
The Volume-Averaging Technique (VAT) is used to derive continuum
equations for multiphase systems. The purpose of the VAT theory is to reduce
the complexity of geometrical structures such as heat exchangers, paper sheets
and other dense structured systems. The method focuses on moving the
attention from establishing valid equations in the pore-space, to using the porescale characteristics to derive volume-averaged equations which are spatially
smoothed over in the entire domain. Upon volume-averaging on the pore-scale
level, integrals involving micro-scale variations of the averaged variable arise.
These integral equations present a closure problem in which a scheme of how
to obtain volume-averaged expressions of the local dependant variable of the
integral equations is essential. Much of the early work in developing the VAT
theory was conducted during the 1960s [8-10]. These papers present a rigorous
mathematical foundation for the VAT. The development of the theory of VAT
and its application in various cases is presented by Whitaker [11]. The
implementation of the VAT has been extended to analysing highly porous and
heterogeneous structures in which turbulent effects need to be considered [12-14].
The principles of volume-averaging are applicable on single and multiphase
systems consisting of different fluids. In the case of modelling multiphase
systems, an important issue is establishing the correct mathematical descriptions
of boundary surfaces that separate the phases [15-18].
Volume-averaging theories have been recently applied to heat exchanger
analysis. Based on the fact that overall heat exchangers comprise a dense
structure of an array of fins and tubes, the heat exchangers are viewed as
porous media in which the specific geometry of the original structure is
replaced with averaged properties of the flow process. By using the VAT
theory, the momentum and energy equations can be solved for several types of
heat exchanger configurations. Major contributions in VAT theory applied on
heat exchangers have been established [19-26]. These papers have, in general,
contributed to establishing fast running computational algorithms based on the
VAT to simulate airflow through mostly an aluminium chip heat sink with a
staggered array of pin-fin configuration and fin-tube heat exchangers. They
handle the closure problem with two essentially different approaches. Much of
these studies used empirical relations based on the experimental work of drag
and heat transfer coefficients in order to handle closure problems from upscaling at the pore-scale level to the macro-level [19-21]. The other approach is
that CFD simulations are used in their Representative Elementary Volumes
(REV) in order to obtain solutions for their integral terms, which represented
local heat transfer and local pressure drag [22-26]. The method enabled them to
investigate possible geometric improvements to achieve higher thermal
effectiveness. Based on the fact that these days, empirical relations from
experimental work and CFD calculations are used to handle closure problems;
it is essential that the applied theory is compared to experimental data regarding
the flow process.
Background, Industrial Application Fields
This chapter presents the application fields encountered in this thesis, which are
the heat pump tumble dryer and the vacuum dewatering process in a paper
machine. Certainly, CFD implementation in design processes and establishing
time efficient simulations in systems regarded as porous media are not limited
to these fields, as these aspects have become more crucial to companies in
3.1 Heat pump tumble dryer
Household appliance products constitute one of the many energy demanding
markets. They consume almost one third of all of the electricity produced by
IEA (International Energy Agency) member countries [27]. The tumble dryer is
one of several energy demanding household appliance products in, among
other places, residential homes. A report by ETSAP [28] presents some
interesting data. They state that in the year 2009, approximately 79% of US
households owned a laundry dryer, and that in the EU there is an annual sale of
approximately 30,000 units. These units mostly consist of electrical dryers. As
there is a general growth in population, the demand for household appliance
products is also increasing. Hence, it is crucial to reduce the energy demand for
such products and to make continuous efforts to increase their efficiency.
The heat pump tumble dryer is a new concept in the household appliance
industry. The working medium of the tumble dryer is transported in a closed
internal duct system, see Fig (3). The humid air is transported from the drum
into a heat exchanger unit, where the evaporated water is condensed in the first
heat exchanger it meets with. In the second exchanger, the air is heated again
before it is transported back to the drum. The energy of condensation is utilised
as a low-temperature energy source, which the working medium in the heat
pump absorbs. Compared to its predecessors, the heat pump dryer is
extensively more energy efficient. As a result of it, the dryer only uses half of
the electricity of a traditional condensing dryer [29]. Moreover, the life cost cycle
of the heat pump dryer is more profitable than the other drying concepts,
making it a sustainable approach towards the drying of clothes [30].
Fig. (3). Components of the internal duct system of a heat pump tumble dryer.
The analysis of the airflow behaviour has yet to be studied or published to a
further extent. Since the production and the manufacturing of tumble dryers are
continuously increasing, analysis of the airflow could provide some motivating
results in terms of reducing pressure drop and noise pollution.
3.2 Papermaking
Papermaking is the process of producing paper, which today is mostly used for
printing, household usage and packaging. The concept of papermaking dates
back to 2000 years ago when the Chinese collected various plants such as
mulberry bark and hemps and separated the fibers onto old rags which they laid
in the sun to dry the web of the fibers, which resulted in a paper sheet. During
the timeline of papermaking, the method evolved and spread to the Islamic
world where the process of papermaking was refined, as machinery was
designed for bulk manufacturing. Papermaking reached Europe in the late 11th
century where paper mills were introduced in countries such as Spain and
France, to name a few. However, in conjunction with the industrial revolution,
modern papermaking took place, which catapulted the production of paper
worldwide [31].
3.2.1 Paper machine
Modern papermaking took place in the early 19th century with the development
of the Fourdinier machine. The concept of producing paper in this machine
differs from previous methods because the paper is produced in continuous
sheets rather than being produced in individual sheets.
Fig. (4). Water removal in a Fourdinier paper machine
The modern paper machine consists of three main operational sections. The
first stage is the forming section; water is removed by gravity and vacuum
filtration is applied at various pressure levels. The second stage is the press
section; water is pressed out of the web through mechanical pressing in several
press nips. The drying section is the final stage where the majority of the
remaining water is removed in a series of steam heated cylinders. The fiber
suspension enters through the head box in the forming section, where the
dryness is approximately 1%. Most of the water in the fiber suspension is
removed in the forming section, and the dryness of the paper is generally at a
level of 20%. At the end of the press section, the dryness is roughly at 40% and
the final dryness is reached in the drying section, which is approximately 95%.
There is a high demand of energy for paper machines during the production of
paper. In the year of 2012, 8,525 paper machines were in operation and the
average annual electrical use for one paper machine was 140 TJ [33]. The ratio of
this demand is approximately 20% for electrical energy input to the vacuum
system. The thermal drying section considerably exceeds the other dewatering
stages in terms of energy use. It was reported that 80% of the total energy used
in a paper machine is steam used for the dryers [34]. Hence, increasing the
incoming dryness to the thermal stage could lead to a major reduction of the
production cost.
Many attempts on further developing the process of papermaking have
involved optimising control systems for various components and sections of a
paper machine [34-36]. These studies implemented control systems which
improved wet end capabilities in terms of improving the consistency of the
formation of the paper sheet and reduction of the unpredictability of the wire
retention. Moreover, they were able to reduce specific steam consumption on
the dryers by 10%. Bhutani et al. [37] developed a solution for a paper machine
steam energy fingerprint. They investigated a dryer section of a paper machine
with a potential steam savings of 10-15%, which was achieved by optimising
the control set points for a steam dryer group.
3.2.2 Vacuum dewatering
The process of applying vacuum in several suction boxes occurs in the forming
section. The forming section is generally divided into two zones. In the first
zone, the diluted fiber suspension is sprayed onto the forming wire from a head
box, whereafter water is removed by gravity and low vacuum levels are initiated
by dewatering elements such as foils. The second zone comprises suction
boxes, generally placed under the forming wire. These boxes apply higher
vacuum pulses at levels ranging from 15 to 65 kPa.
Most of the research conducted on vacuum dewatering has either been done by
using laboratory equipment or by using pilot paper machines. The process
parameters that are usually investigated are the influence on the dryness by the
applied vacuum, the dwell time and the basis weight [38-44]. The mechanisms
involved in vacuum dewatering have been described as web compression,
displacement of water by air and rewetting from the forming wire [42,45,46]. Even
though there are established design data based on experimental work done on
vacuum dewatering, there is still a lack of validated CFD-models predicting
dewatering rates in relation to dwell times for various basis weights of paper
sheets. Capturing all dewatering mechanisms is challenging, considering the
movement of fibers and the inclusion of capillary forces to capture the
rewetting phenomena. However, conducting experimental analysis on large
scale paper machines requires costly resources. Establishing numerical models
would yield a more cost efficient process for analysis of process parameters
such as the basis weight of the paper sheet, testing various vacuum levels and
duration of pulses, analysing the influence of different forming wire
configurations, etc.
Forming wire
The forming wire is an essential component in the process of paper production.
The wire is a 3-dimensional woven matrix which consists of orthogonal crossdirection filaments in several layers, see Fig. (5). During the process, the wet
pulp is dewatered while on the forming wire, which in turn yields the formed
paper sheet. Hence, the capability of the forming wire is significant in terms of
achieving a more energy efficient dewatering process; more economic
sustainability, by reducing the tear of the wire material; improvement in the end
quality of the paper sheet.
Fig. (5) . A CAD-model illustrating the arrangement of the forming wire.
There has been research done regarding the influence of the forming wire on
the dewatering process. Granevald et al. [47] studied the influence of ten
different forming wires on the sheets’ solid content and the dewatering rate
during vacuum dewatering of low grammage sheets. They concluded that
caliper, void volume and air permeability are three important parameters. The
forming wire consists of yarns with a diameter typically around 100-200 µm on
the paper side, and a pitch of 200-400 µm. The geometric configuration is far
Albany International Corporation
coarser than the paper, which should not contribute much to the total flow
resistance. Still, there is a reason to investigate the influence of the contact
surface between the paper and the forming wire as a result of the forming and
packing of the fibers on the pore space during the high pressure vacuum
process. In a review article by Hubbe et al. [48], they addressed the issue
involving the interaction between fibers and the forming wire. Aside from
improving the performance of the forming wire through analysis of flow
properties, surface treatment was conducted in terms of a heating compaction
process that flattened out the top surface yarns on the paper side of the
forming wire [49]. This reduced the fabric caliper and internal void volume by 515% and, in turn, improved vacuum dewatering efficiency.
Theory of Fluid Mechanics
The mathematics that describes flow behaviour is challenging, yet we encounter
complex flow patterns in everyday life. Water, for instance, flows out of a tap,
smoke rises up a chimney, the irregular motion of a leaf falling from the tree
and the flow distribution in a lecture room are only a few of an innumerable
amount of events that involve complex flow patterns.
It is difficult to establish when the study of fluid motion first took place, due to
the art of documentation being relatively new to mankind. The development of
hydraulics was purely empirical, as our ancestors built farms where the
utilisation and transportation of water were crucial. The discovery of
constructed irrigation canals dating from 4000 years B.C. was made in ancient
Egypt and Mesopotamia. More advanced water systems for transportation and
storage were constructed in old civilisations such as Jerusalem, Greece and,
especially, the Roman Empire. The evolution of the ship is a result of empirical
studies of fluid mechanics. Mankind first developed simple boats out of logs
and moved towards advanced ships with sails, which allowed them to navigate
the oceans. The Phoenicians and Egyptians are known for having built
excellent ships.
During the second half of the 15th century, a polymath named Leonardo Da
Vinci was one of the first scientists to visualise and document fluid flow in
detail. He wrote extensive descriptions of the movement of water in the form
of eddies, water waves, free jets and falling water, to name a few. His work
paved the way for future research in the fields of hydraulics, as well as
hydrodynamics. In the late 17th century, Isaac Newton tried to quantify fluid
flow through the elementary Newtonian physical equations. This includes the
concepts of the Newtonian viscosity and reciprocity principle. His famous
second law:
, is the foundation for the deduction of the Navier-Stokes
(NS) equations.
Worlds of flow – A history of hydrodynamics from the Bernoullis to Prandtl. New York, 2005
Fig. (6) and (7) : Studies of water passing a solid object and water falling into still water by
Da Vinci.
In the mid-18th century, Daniel Bernoulli contributed significantly to the field
of hydrodynamics by trying to mathematically describe the motion of fluids. In
his treatise, Hydrodynamica, he acknowledged the properties of basic importance
with regard to fluid flow such as pressure, density and velocity. His famous
Bernoulli’s principle was crucial in the early studies of aerodynamics. During
the time of Bernoulli, Leonard Euler proposed the so-called Euler’s equation,
which describes conservation of momentum and mass for an inviscid fluid.
George Gabriel Stokes, who introduced viscous transport to the equations, and
together with Claude Louis Marie Henry Navier, conducted further
developments on Euler’s equations, These equations form the basis of modern
day CFD usage.6
4.1 Pressure in fluid flow
The concept of pressure is an integral part of the field of fluid dynamics. The
unit for pressure is Pascal (Pa), which equals force, (N), divided by unit area
(m2). Hence, the definition of the term pressure is that of a force that is
perpendicularly applied to the surface of an object. The term is a scalar
quantity, as it relates the normal vector of the surface and to the force vector
acting in a normal direction to it. The concept of pressure was introduced in
ancient Greece when Archimedes presented the principle of buoyancy, stating
that force acting on an immersed body is equal to the weight of the water, 2011-01-10, 00:57
5, 2011-01-10, 01:35
6 Worlds of flow – A history of hydrodynamics from the Bernoullis to Prandtl. New York, 2005
volume it displaces. The principle is based on a static system in which the
pressure acts on a non-moving object.
Pressure can be identified at every point in a body of fluid, regardless of the
fluid being in motion or not. The first equations coupling velocity to pressure in
fluid motion equations were introduced by Daniel Bernoulli in his treatise
Hydrodynamica. They are described in the so-called Bernoulli's law. He used the
concept of water decanting in a vertical vessel as a result of the efflux of water
in an attached tube in the lower section. He stated that the acceleration of the
water in the intersection of the vessel and the tube is equal to the loss of the
fluid's potential energy. He put it in words that the exerted pressure by the fluid
on the walls of the vessel is equal to the difference between the hydrostatic
pressure and dynamic pressure. This statement led to what is known today as
Bernoulli's equation:
Later on, Daniel's father, Johann Bernoulli, stated that the pressure of the fluid
in the vessel is a result of adjacent fluid parts exerted on one another, which he
called internal pressure. This pressure is today better known as the static
pressure of the fluid, and it is represented by the first term on the left hand side
of Eq. (1). The second term to the left of Eq. (1) is often called the datum of a
pipe or channel and the third is the dynamic pressure, which is called kinetic
energy, if Eq. (1) is interpreted in energy units. Bernoulli's statement is that the
energy density is constant following a streamline in the flow, which is the sum
of all terms in Eq. (1). Moreover, friction losses due to viscous stress are
When studying fluid flow through a pipe or channel, a similar approach to
determine the relation between the velocity and pressure is to apply the
conservation of energy. If the frictional losses were disregarded, it would result
in Bernoulli's equation. With a presented control volume of a fluid channel with
a steady state flow at hand, however, the conservation theorem can be set up in
different forms of conservative units such as mass, momentum or mechanical
energy. The continuum principle is one of the central assumptions in this field.
Here, the discrete nature of matter is overlooked, given the fact that the length
scale of the macroscopic matter is large, compared to the atomic discrete length
scales. Thus, the matter is treated as a continuum, which indicates that the
quantities of the matter are continuously differentiable. In 1750, Leonhard
Euler claimed that Newton’s second law of motion applied to infinitesimal
bodies was the true basis of continuum mechanics. He stated that the
acceleration of a fluid element depends on the combined effect of the pressure
gradient and external forces, such as gravity. Based on Newton’s second law,
Euler obtained what is known as one of the first PDEs for fluid motion. 7
Eq. (2) is written for a steady state process, as the time derivative is disregarded.
To study the conservation of an arbitrary control volume, the divergence
theorem by Johann Carl Friedrich Gauss is used to clarify the balance of the
equation. The divergence theorem states that the outward flux of a vector field
through the surfaces of a closed control volume is equal to the volume integral
of the divergence of the vector field in the region.[50]
The term on the left hand side of Eq. (3) represents the divergence of the
vector field inside the control volume, which represents the sum of all sources
and sinks. The term on the right hand side of Eq. (3) represents the vector field
acting in the normal direction at the surface of the control volume. Hence, this
theorem is a conservation law, and it is applicable in other fields such as
electromagnetism and quantum mechanics.
Euler’s momentum equation may be applied to an arbitrary control volume in
which the sum of all forces on the flux boundaries is determined. The force
that represents the vector field described in Eq. (3) is the parameter of interest
in the control volume.
Worlds of flow – A history of hydrodynamics from the Bernoullis to Prandtl. New York, 2005
Fig. (8). Momentum flow balance over an arbitrary control volume.
Applying force balance in terms of momentum fluxes and pressure forces on
the boundaries of a control volume indicate the state at each end point, see Eq.
According to the divergence theorem, the sum of all forces that are exerted on
the fluid may act like sinks in the region.
4.2 Navier-Stokes equations
The most common PDE in fluid flow analysis are the NS equations. The
equations are nonlinear PDE that state that the momentum change rate per
unit volume of a fluid element (term on the left hand side of Eq. (6))
corresponds to pressure (first term on the right hand side) and viscous forces
(second term on the right hand side) acting on the element boundaries. The
continuity equation states that mass flux entering and leaving the control
volume is equal to the change in mass, see Eq. (5).
In some cases, gravitational forces need to be introduced. When dealing with
air, this term is often negligible. When dealing with the incompressible form of
the NS equations, the term is constant. In the view of flows with significant
temperature variation, buoyancy effects need to be considered. Moreover, the
compressibility in the NS equations needs to be considered when dealing with
high velocities (Mach number > 0.3). The NS equations are similar to Euler’s
momentum equation in Eq. (2), except for the added term of viscous forces.
The indices and in Eq. (2), (4),and (6) represent three coordinate equations
in which and may take on the values of 1, 2 or 3. If we want to write the NS
equations in the x-coordinate, we set to 1. Since the index appears twice in
the convective and diffusive term, we have to apply the summation rule due to
the fact that the index is regarded as a so-called dummy index.
Further contributions to the boundary layer theory and turbulence were made
by Ludwig Prandtl during the first half of the 20th century. While Prandtl was
working in die Maschinenfabrik Augsburg-Nürnberg, he attempted to improve
a suction device for the removal of shavings. During the course of the project,
he realised that the pressure rise expected in a divergent tube failed to occur
because the lines of the flow tended to separate from the walls. On the basis of
this observation, Prandtl founded his well-known boundary-layer approach to
resistance in viscous fluids. Prandtl stated that a fluid with a relatively low
viscosity, such as water or air, is inviscid in a flow domain; except near the wall
where the fluid adheres to the solid wall. This region is called the boundary
layer and it is based on the assumption of the no-slip condition. Furthermore,
he put forth several approximations for the boundary layer through scale
analysis, in which he determined that for a 2-dimensional flow, the streamwise
velocity is larger than the spanwise. Yet, the derivative (change) in the spanwise
direction is larger. Moreover, he determined that the change in pressure over
the boundary layer thickness is negligible.8
Worlds of flow – A history of hydrodynamics from the Bernoullis to Prandtl. New York, 2005
4.3 Accounting for turbulence
Already in the late 18th century, Daniel Bernoulli and his father shared a
preconceived notion about mechanical losses that was later put forth by
Leibnitz. The notation states that losses of mechanical energy in fluid motion
occur in the interaction of small-scale motion. Bernoulli's equations exclude
fluid resistance, which was judged by Bernoulli to be beyond the grasp of
mathematics at the time.9
Turbulence is a common occurrence in practical fluid flow. Turbulence is
characterised by randomness or chaotic behaviour of the flow in which
turbulent eddies occur at different velocity, length and time scales. High values
of momentum diffusion occur in turbulent flow, due to effective mixing caused
by the eddy motion. The length scale of the largest eddies in a flow process is
of the same order as the flow geometry. These eddies are created in presence of
a mean velocity gradient, hence kinetic energy in the mean flow is transferred to
the eddy motion of the largest scales. The transfer of kinetic energy continues
to a smaller eddy and so on. At the smallest scale, the eddy dissipates to thermal
energy, as the kinetic energy in the eddy motion is not able to overcome the
viscous forces in the fluid. The process in which the kinetic energy is
transferred from the largest eddies to the dissipative ones is called the cascade
process. The dissipative scales were put forth by Russian mathematician Andrey
Nikolaevich Kolmogorov, they are famously known as Kolmogorov scales.
In theory, the NS equations describe laminar as well as turbulent flow
conditions. With increasing Reynolds number however, disturbances in the
flow are created due to the development of fluctuations in the velocity and
pressure fields. Hence, in the presence of turbulence, it would require a great
amount of mesh elements to resolve all the turbulent scales in spatial
coordinates and a fine resolution in time, given the fact that turbulent flow is
unsteady. Direct numerical simulation is equivalent to applying the NS
equations without an additional turbulence model to resolve a flow process.
Such simulations are generally not feasible for standard computer usage, as the
mesh grid easily exceeds millions of elements for relatively small Reynolds
numbers. This issue has paved the way for extensive research on developing
turbulence models.
Worlds of flow – A history of hydrodynamics from the Bernoullis to Prandtl. New York, 2005
4.4 Two-phase flow
Systems involving simultaneous fluids, e.g. a gas and liquid phase that are
immiscible, are referred to as a two-phase flow system. Other classifications of
two-phase flow are liquid-solid and gas-solid. In practical systems, the twophase flow phenomena appear in a wide variety of engineering conditions, e.g.
dewatering during papermaking. Key areas have benefited from two-phase flow
studies such as pump cavitation, in which vapour pressure near regions of the
pump or propeller can be detected. In industrial systems such as power plants,
two-phase flow phenomena occur in combustion and boiling processes in
which the behaviour of the flow, heat transfer and pressure evolution are
significantly different from single-phase systems.
In two-phase flow systems with an interface boundary, phenomena are
encountered which are caused by the existence of surface tension on the
interface boundary of the fluids. Cohesive and adhesive forces arise from
intermolecular action close to the surface. These forces arise at the surface of
the fluid due to unsymmetrical net force balance, thus causing the molecules to
contract to a minimal area of the liquid surface, e.g. formation of droplets. In
other words, surface tension affects the dynamics of the flow process as there is
a change in the boundary condition of the interface. A classical description of
surface tension forces is presented in Fig. (9), which depicts a capillary tube
Fig. (9). Schematic description of surface tension forces in a capillary tube.
The equilibrium of forces is the surface tension forces due to curvature
balanced by the weight of the liquid. The surface tension is described in
differential form according to Eq. (7) [51].
In Eq. (7), the left-hand side is the force acting on a surface at a location, based
on the cross product of the curve element and the surface normal determining
the position and direction of the force. In the case of the capillary tube, the
body force is cancelled out by surface tension, according to Eq. (8).
The significance of the capillary force could roughly be determined by the
dimensionless Capillary (Ca) number or the Bond (Bo) number; see Eqs. (9)
and (10) [52,53].
The Ca-number describes the ratio of viscous forces to capillary forces, whereas
the Bo-number presents the ratio of gravity forces to capillary forces. The term
presents the filtration velocity,
is the density difference in the two fluids,
is the fluid viscosity of the liquid, is the surface tension coefficient, is the
gravitational acceleration and is the characteristic dimension of the pore size.
Historically, the two-phase flow phenomena have extensively been studied in
various fields through experimental and empirical analysis, due to the complex
nature of the flow process. These studies are also limited as a result of their
complexity, which has led to the implementation of numerical analysis as a tool
for understanding these processes.
4.5 Flow in porous media
The concept of porous media flow is widely encountered in various fields such
as filtration theory, petroleum engineering, soil mechanics, etc. A porous
medium is characterised by subdividing the total volume of the system into a
solid matrix and a pore space. The solid matrix is constituted by an
interconnected pattern of solid materials, creating a dense structure. Such
structures include e.g. a packed bed of solid grains, an entanglement of fibers
and a connected pin-fin arrangement from a heat exchanger. Typically porous
materials are, e.g. rocks, wood, paper, bones and ceramics, to name a few.
These days, the flow in porous media has emerged as a separate field of study,
due to its wide interest and application field.
4.5.1 Fluids and porous media as continua
The concept of continuum mechanics is to consider a modelled material as
continuous mass rather than discrete particles. By definition, when considering
a matter as a continuum, the body can be continually subdivided into
infinitesimal elements in which the properties of the body are the same as the
bulk material. All materials are, however, composed of molecules which
discretely connect atoms and, in turn, fail to meet the criteria of being
continuous in space. Modelling fluid mechanics based on the Lagrangian
perspective would result in describing fluid behaviour on a molecular scale in
order to track the behaviour of each molecule, which would not be feasible due
to the complex nature of the fluid. The fundamental assumption in continuum
mechanics is to model materials on length scales far greater than the interatomic distances in order for the statement to be valid. Thus, we must diverge
from the Lagrangian concept and involve microscopic scale effects in terms of
averaging. The same concept is applicable for porous structures, as the aim is to
make a transition from modelling constitutive variables in the pore space to a
continuum of macroscopic field variables. In turn, the detailed structure of the
pore geometry is eliminated.
4.5.2 Flow representative volume
Numerical analysis of flow in porous materials in practical environmental
systems is not feasible when a detailed description of the pore geometry in the
scale of micrometres is the objective. Thus, when choosing a macroscopic scale,
in which the details of the flow process are ignored, the average behaviour of
the fluid in the porous medium is modelled. The macroscopic behaviour of the
flow is determined by volume-averaging the constitutional variables of the mass
and momentum equations over a so-called Flow Representative Volume (FRV).
The FRV is viewed as a sample of a porous medium in which the size of the
FRV determines the level of variation in the solid matrix structure of the
porous medium, upon which the volume-averaging is executed. In other
literature regarding random particle generation with, e.g. statistical models on
samples, they refer to these volumes as a representative elementary volume. A
porous medium could be categorised as an ordered or disordered medium, see
Fig. (10).
Fig. (10) Examples of FRV-models representing a characteristic structure of the porous
medium. In (a), the FRV is chosen for an ordered structure of a pin-fin arrangement in a
heat exchanger; and in (b) the FRV represents a disordered arrangement of a fibrous
structure representing a paper sheet.
The structure of a classical pin-fin arrangement could be regarded as an ordered
porous media due to a repeated identical structure in the streamwise and
spanwise direction, see Fig. (10a). Choosing an appropriate FRV is to find the
minimum repetitive volume in the sample. Depending on the size of the heat
exchanger, the FRV should be extended in order to include the effect of
entrance regions and re-circulation effects near the outlet.
In Fig (10b), the FRV consists of a disordered porous medium where
cylindrical elements have been randomly distributed over a sample volume,
which in this case represents a paper sheet. The choice of FRV is based on
different criteria than for an ordered structure such as the heat exchanger. Due
to the irregular shape and displacement of solid particles in the sample volume,
there is a distribution of porosities within the sample. Selecting a small size for
the FRV would increase the heterogeneity of volume fractions of each phase.
Increasing the size of the FRV would, however, smooth out local
heterogeneities as the scale of the sample far exceeds the particle scale.
4.5.3 Reynolds number
In the theory of fluid mechanics, a key dimensionless quantity, which describes
the unstable nature of the flow is called Reynolds (Re) number, named after the
Irish engineer Osborne Reynolds. The Re-number represents the ratio of
inertial forces to viscous forces, see Eq. (11).
The quantity is used to characterise different flow regimes within the same fluid
and system. With dominating viscous forces, the flow process is laminar and
with increasing inertial forces, the flow is trending towards a turbulent nature.
In porous media, a modified Re-number is used to determine different flow
regimes in which the velocity term and the characteristic length scale are
modified to the particle size characteristics of the porous medium. In a system
containing solid particles, the so-called interstitial velocity, which is the velocity
in the pore-space, is the appropriate term. This term is related to the superficial
velocity with the porosity of the medium, in order to preserve fluid continuity,
see Eq. (12).
The term
denotes superficial velocity, which is viewed as the velocity in the
medium removed from its solid particles, and is the porosity of the medium
which is the ratio of pore space volume (void) to the total volume, see Eq. (13)
and Fig. (11).
Fig. (11). Representation of flow through a column of unidirectional circular fibers.
The characteristic length scale of a porous material with a packed bed of solid
particles was deduced by the Austrian physicist Josef Kozeny, and it is the
volume of the pore-space divided by the specific open area over which the fluid
must flow, see Eq. (14).
The term
represents the specific surface area, which in the case of circular
fibers is the ratio 4/D. The characteristic length scale along with Eq. (12) is
implemented in Eq. (11), in order to achieve the so-called particle Re-number.
The particle Re-number represents the ratio of inertial to viscous forces, just as
in the original expression. This expression is viewed as a bed average quantity
and will accurately represent a flow regime with fairly uniform flow patterns
and geometric properties. Due to the complex configuration of the solid matrix,
locally the flow will exhibit various conditions such as laminar with low inertia
and high inertia and turbulence with unsteady flow patterns. However, with a
large bed geometry compared to the particle size of the porous medium, the
effect of these local heterogeneities should be reduced. This is further evaluated
in chapter 5.2.3.
The threshold for inertial effects for the particle Re-number is extensively lower
(Re > 10) in porous media than regular pipe flow systems [54-57]. There are other
studies that claim that a weak inertia term exists in Re-numbers close to unity
[58,59]. Strong inertia, which can be viewed as a transitional flow regime, exists in
the range of Re-numbers at 10 to 200, which are then followed by an unstable
turbulent flow. However, these critical values are still not clearly defined, due to
difficulties in observing fluid behaviour.
4.5.4 Viscous and inertial contributions in porous media
One of the first scientists to study fluid flow in a porous medium was the
French engineer Henry Darcy. His famous Darcy’s law was used to solve
problems such as water transport in an aquifer, or oil migration to a well. These
processes of fluid transport are characterised by low flow velocity and a dense
structure of pores in the material, i.e. low porosity. Darcy’s law is established
based on experimental analysis at very low Re-numbers and resembles the NS
equations excluded from the advective term, see Eq. (16).
The advective term representing inertial forces is disregarded, which yields a
linearisation of the NS equations to Stokes flow, which is presented in steady
state without body forces. Assuming that viscous resistance is linear to the
velocity, the forces are characterised with the porous properties of the medium,
i.e. the porosity and permeability. A homogenisation of the pore structure and
the fluid into one single medium is a common approach when using Darcy’s
law, in which the pressure gradient is the major driving force, Eq. (17).
〈 〉
In this equation, is the permeability of the porous medium and it represents
the resistance to flow of the medium. Depending on the isotropic nature of the
porous material, the permeability could be defined as a second-order tensor,
indicating anisotropic resistance or a scalar term indicating identical resistance
spatially in the material.
In the second case the indices are identical, which results in the summation of
the diagonal term in Eq. (18), see Eq. (19).
Along with the pressure gradient, gravitation is a driving force of the flow.. A
rigorous mathematical derivation of the Darcy’s law is presented in a theoretical
description of how the permeability tensor is determined [60].
An extension of Darcy’s law was conducted in which the transitional flow
between boundaries separating an open area to a porous area is taken into
consideration [61]. Darcy’s law is extended with a term associated with drag
forces experienced by the fluid flowing in the permeable region of the porous
medium, see Eq. (20).
In porous media, where Re-numbers exceed the value of creeping flow
conditions (Re > 1), weak inertial effects should be considered as momentum
losses are influenced by non-linear effects. The deviation from Darcy’s law has
been observed, based on experimental analysis and through empiricism;
researchers have extended Darcy’s law with correction terms including inertia in
the flow process. The most notable formulation is the one presented by the
Austrian engineer Philip Forchheimer in 1901, which today is known as the
Forchheimer equation and is presented in Eq. (21).
are vectors that denote the volume force (N/m3) and velocity field
(m/s), respectively. The
coefficients represent viscous and inertial
resistance, respectively. The high velocity inertial effects are included by the
non-linear term representing the kinetic energy of the fluid. The viscous
resistance coefficient is related to a permeability second-order tensor in Eq.
(18), for anisotropic porous structures. The inertial resistance coefficient has
been the foundation of a research area, which is based on extensive
experimental studies in terms of empirically determining the validity of the
Forchheimer equation. In recent times, however, various theoretical approaches
have been used to mathematically derive the equation from its first principles. A
thorough presentation is given of how Darcy’s law, along with the Forchheimer
correction terms are derived through a volume-averaging method [62]. Along
with the VAT theory, homogenisation theories are used to derive
Forchheimer’s equation [63,64].
Non-Darcy flow has been the target of many numerical studies as well in which
the main focus was set on determining the inertial resistance coefficient in both
isotropic and non-isotropic porous media in 2- and 3-dimensional systems [55,6570]. According to Wang et al. [71], the inertial term has a tensorial form if the
porous structure is anisotropic. The Forchheimer equation is valid for low as
well as high Re-numbers as was stated by Andrade et al. [66]. They observe,
however that at a deduced friction factor from the Forchheimer equation
overestimates flow resistance compared to experimental data in range of Renumbers 0.06 to 11.
In terms of characterising inertia in porous media, researchers have proposed
Forchheimer’s equation to empirically account for non-linear effects based on
experimental and literature data [72-79]. In practice, the inertial resistance
coefficient is usually determined by fitting the data in which the aim is to
achieve correct correlations of experimental data for various systems. The most
widely used empirical expression of the inertial term was put forth by Ergun [80]
and is expressed as:
is the dimensionless friction coefficient deduced empirically by Ergun
and could be expressed as:
The model expression established by Ergun satisfies linear and nonlinear flow
regimes at a wide range of Re-numbers with reasonable accuracy. The Ergun
equation has been the foundation of extensive experimental research on flow
through packed beds, both as a validation tool and as a reference model for
which modifications have been conducted in order to assess the validity of the
model and, moreover, to improve the accuracy of the model.
The method described in this thesis is divided into three main categories:
Modelling in COMSOL
Volume forces coupled to single and two-phase flow
Characterisation of flow resistance in various fibrous structures
The first category, (i), is a brief introduction of the simulation program
COMSOL. The second category, (ii), is a description of the methodology of
establishing volume forces as sink terms in the momentum equation. The third
category, (iii), is characterisation and assessment of the reliability of these
volume forces through comparison of our numerical data to classical empirical
formulations of the dimensionless friction coefficient, as well as current
numerical results.
The presented results in this thesis will involve categories (ii) and (iii), as the
focus has been to present the reliability of the volume force terms and assess
the benefit of implementing them in structureless domains regarded as porous
5.1 Modelling in COMSOL
This chapter presents a short
COMSOL MultiPhysics, in which
handling numerical instabilities
Additionally, the two-phase flow
briefly introduced.
introduction to the simulation program
the discretisation method, the generic PDE,
and boundary conditions are addressed.
model called the Level-Set (LS) method is
Finite Element Method
The discretisation method used in COMSOL is the FEM. The development of
the FEM began in the middle of the 20th century, mostly for structural analysis
in civil engineering. The method evolved as it gathered a strong mathematical
foundation and has since been generalised to several fields of applied
mathematics for numerical modelling, such as fluid mechanics.
The method subdivides an object into small finite-size elements. Each element
is described by a number of DOF in which a set of characteristic equations is
solved simultaneously. The finite element approximations of PDEs are found
in the space of linear functionals. This space is called the finite element space. A
brief introduction of simple problems is presented as we discuss polygonal and
planar domains.
The concept of the method is that a known surface is represented by an
approximate surface:
For a one-dimensional problem, the curve is approximated with a linear
), then a
interpolation. If and are the endpoints of an interval,
number of points (nodes) are selected on the interval:
The piecewise linear interpolant ( ) is formed by connecting each node with
a neighbouring node, see Fig. (12).
Fig. (12)
A curve and its piecewise linear interpolant.
Each node is associated with a basis function ( ). In numerical analysis, a
basis function provides an interpolating function of a curve or surface in the
function space, see Eq. (26).
Introduction to the Numerical Analysis of Incompressible Viscous Flows, pp. 18
( )
( )
( )
In two dimensions, the approximation of the function is on a surface and it is
( ), in which (
) is defined on
done by introducing a triangulation,
each triangle.
Fig. (13)
A finite element space formed as a hat-function.
The triangle is a finite element space. For the finite element space,
is a linear
or higher order polynomial on each mesh element, with the value of 1 in node
and 0 in all other nodes. A few basic conditions are required for creating a
triangulation: the triangles are conforming, which indicate that the vertex of one
triangle cannot be positioned on the edge of another triangle; the triangles are
not extensively skewed; no triangle has all three vertices on a part of the
5.1.2 Numerical instabilities
During numerical calculations of e.g. a flow process, approximation errors
generally occur. In cases when the error is amplified, the solution is referred to
numerically instable.
There are techniques applicable for handling numerical instabilities without
having to refine the mesh structure further. In the case of a Galerkin finite
element method (GFEM), which is used in COMSOL, an artificial diffusion
Introduction to the Numerical Analysis of Incompressible Viscous Flows, pp. 19
coefficient can be applied to COMSOL’s generic scalar convection-diffusion
transport equation:
Eq. (27) is a general PDE in COMSOL, which describes transport phenomena
such as heat, mass and momentum transfer. In the case of momentum transfer,
represents the advective velocity vector; represents the viscosity in this
is the transported velocity vector and F is a source term. When
discretising Eq. (27), the Péclet (Pe) number indicates instability when the value
surpasses one.
‖ ‖
The dimensionless quantity Pe was deduced by the French physicist Jean
Claude Eugène Péclet during the first half of the 19th century. The quantity is
defined as the ratio of the rate of advection of a quantity due to the influence of
a fan or pump, to the rate of diffusion of the same quantity due to gradients. In
this case, Eq. (28) indicates that it is preferable that the convective term has a
low value and/or that the size of the mesh cell, ( ), is small in order to keep Pe
< 1. Generally speaking, simulation of flow with high velocity requires a dense
mesh structure. Moreover, when treating fluids with a high viscosity, such as
oil, it damps the effect of oscillations due to the fact that the coefficient is in
the denominator of Eq. (28). The artificial diffusion is implemented by adding a
term to in Eq. (27).
‖ ‖
The tuning parameter
transport equation is:
has its default value at 0.5. Hence, the modified
in which the new Pe number is:
‖ ‖
‖ ‖
The new Pe number will not exceed one if ‖ ‖ approaches infinity. The tuning
parameter, however, should be set as low as possible, given the fact that the
new modified transport equation could present solutions that deviate to far
from the original problem.12
5.1.3 Boundary conditions
In COMSOL, there are several predefined boundary conditions for fluid flow,
heat and mass transport. These conditions are applied by the user, depending
on the physical conditions surrounding the computational domain. The
boundaries are characterised as exterior, if they encompass the computational
domain. The interior boundary condition is a dividing interface between two
subdomains for the entire computational model, see Fig. (14).
Fig. (14). Representation of an exterior and interior boundary.
For interior boundary conditions, COMSOL ensures continuity of fluxes across
the interfaces in most cases.
In CFD, there are two basic types of boundary conditions:
 The Dirichlet boundary condition
 The Neumann boundary condition
COMSOL Multiphysics documentation
Classical boundary conditions such as the no-slip and constant pressure are
characterised as Dirichlet conditions, indicating that the constitutional variables
are fixed at the boundary.
The general heat flux condition is a typical Neumann condition, as the
temperature on the wall is not fixed.
Even though a constant heat flux is defined at the boundary, the absolute value
of the temperature may vary spatially.
In COMSOL, the generic notations of the boundary types are:
The coefficient
represents a physical quantity such as temperature,
concentration and/or velocity. Eq. (34) represents the Dirichlet condition,
which specifies a constant value on the boundary of the domain. Hence, the
is applied at the boundary. The Neumann condition specifies the
normal derivative of the quantity at the boundary, see Eq. (35). The term
of Eq. (35), represents the diffusive term,
is the convective term, and
represents a source term. The weight coefficient is an
-matrix, where
is the number of dependant variables.13
5.1.4 Level-Set method
Along with solving the NS equations in flow process, a suitable tracking
method is required in order to accurately determine the location of the interface
separating two immiscible fluids. There are several techniques which are used to
COMSOL Multiphysics documentation
trace the interface of two-phase flows. The two most common techniques are
based on the Eulerian and Lagrangian approaches. The Lagrangian method is
suitable for “simple” flows, as the interface surface is explicitly tracked by
marking the interface with particles used for tracking. In complex flow
processes, many tracking particles are required to adequately describe the shape
of the interface, which increase the probability of numerical convergence issues.
Therefore, the method used in this study is based on the Eulerian approach.
The Eulerian based LS method is applied, which numerically tracks the
interface in a stationary Eulerian grid. The grid of the interface is, however,
separately configured and moves along the interface during the simulation
process. The LS function is used to locate, in this case the air–water interface,
in which the function takes a positive or negative value depending on which
side of the interface is viewed:
(⃗ )
where ( ⃗ ) is a scalar quantity named the LS function, and
is the
interface boundary. The LS function is assigned to every grid point in the
domain, thus determining whether the scalar function is in the liquid or gas
phase, see Fig. (15).
Fig. (15). Schematic illustration of a curve representing the interface between two fluid
The interface motion is defined by an advection equation in a Eulerian
where the first term in this PDE is the local time derivative of the LS function,
and the second term represents the advection of the interface. The LS function
is a smooth function and is assigned as a distance function, according to Eq.
| ( ⃗ )|
(| ⃑
⃑ |) for all ⃑
where ( ⃗ )
when ⃑ is equals the space coordinate at the interface
boundary ( ⃑ ). The signed distance function is approximated numerically, in
order to achieve smoothness of the solution. This is achieved by a first-order
accurate approximation using a Heaviside function:
( ( ⃗ ))
( ⃗)
( ⃗)
( ⃗)
( ⃗)
( ⃗)
is a parameter that determines the thickness of the numerically
smeared interface.
As Eq. (37) is solved during a simulation process, the motion of the fluid–fluid
interface changes the LS function, and causes the regulated interface thickness
to increase from a designated signed distance function. This leads to inaccurate
solutions of local properties at the interface. The achievement of a constant
interface thickness minimises the numerical error, due to smearing from
diffusive effects. Consequently, Eq. (37) is extended with a re-initialisation
| )
is the re-initialisation parameter and
equals ( ( ⃗ )). Both the
density and the viscosity are functions of the LS function, which spatially
smooth the variation of the material properties along the interface thickness.
The interface boundary moves with a propagation velocity normal to the
surface, see Fig. (15). This velocity is a function of the fluid velocity, the
curvature and the normal direction of the boundary. It is therefore, important
to accurately model the advancement of the interface boundary. The normal
vector and the curvature at the air/water interface are determined with the LS
Here, the spatial derivative points in the direction of an increasing value of the
LS function. The curvature is calculated from the divergence of the normal unit
| )
Here, the parameter is positive for convex regions and negative for concave
regions. The curvature term is important in numerical simulations of two-phase
flows. When the interface boundary is propagating at a constant speed, it may
yield singularities at several locations of the boundary. In these regions, the LS
function is not differentiable and the evolution of the boundary is not clear.
The curvature term is used to smooth and reshape the moving front of the
boundary, preventing such occurrences.
5.2 Volume forces coupled to single and two-phase flow
The aim of this chapter is to present a method to establish time efficient
simulations. The concept is to reduce the geometrical complexity of systems by
replacing the internal structure with sink terms in the momentum equation. The
chapter consists of: addressing various types porous structures, e.g. a heat
exchanger and a paper sheet; modelling surface tension; assessment of box size,
mesh configuration and sensitivity of the amount of numerical sample points;
estimation of Forchheimer coefficients from numerical data set; establishing
and implementing volume forces in structureless models.
5.2.1 Pore-scale simulations
Both a dense structured heat exchanger component and a paper sheet fall under
the category of a porous medium. However, there are several characteristics of
each component, which set them apart. A classic fin-tube heat exchanger is
ordered in its arrangement and its solid matrix has a repetitive appearance in the
system; whereas the paper is composed by irregular shapes of fibers which are
spatially disordered in the system, Figs. (16), (17) and (18).
Fig. (16) and (17). A cross section of a tube-fin heat exchanger and a closer look at the
arrangement of fins and tube bundles.
Fig. (18). Microscopic view of aligned fibers with various shapes.
As it was stated in chapter 4.5.1, it is not feasible to model constitutive variables
at a pore-scale level if the entire system is to be considered. Creating a mesh
grid for a geometrically compact system generally results in a dense grid
arrangement which requires extensive computer power. The purpose of volume
force implementation in the momentum equation is to bypass a dense mesh
grid arrangement by removing the internal structure.
The method of implementing volume forces in COMSOL is similar to the VAT
theory presented in chapter 2.3. An appropriate FRV is established in the
system in which a direct numerical simulation of the NS equations is conducted
at the pore-scale level. The COMSOL notation of the NS equations is symbolic
and includes a body force term , see Eq. (43).
) )]
In flow processes dominated by the pressure gradient, the term is generally
disregarded. However, in the case of modelling the flow process in a dense
structure, the parameter is included as a sink term in the momentum equation
to account for the resistance force due to the presence of the porous medium.
To put it in perspective with the VAT theory, a closure model is used to
determine in the momentum equation.
5.2.2 Modelling surface tension
In two-phase models, a key parameter needs to be considered in the
momentum equation, which is the contribution of capillary forces. In
COMSOL, surface tension forces are defined according to Eq. (44).
( (
) )
Here, is the Dirac delta function,
is the unit matrix, and is a surface
tension coefficient. When a system involves a moving fluid-fluid interface, the
conditions concerning the solid boundaries are treated by implementing
artificial friction forces in order to prevent numerical difficulties when the fluidfluid interface stagnates at the solid boundary, due to the no-slip condition. The
wetted wall function in COMSOL is applicable at the solid walls to allow for
capillary forces and also to provide greater numerical stability than the no-slip
Eq. (45) indicates a slip condition at the solid walls. In addition, this condition
adds a frictional force, where is the slip length. It is the length in which the
extrapolated tangential velocity is set to zero at the distance from the surface,
see Fig. (19). The slip length is equal to the local mesh element size.
Fig. (19) . Representation of the slip length in the wetted wall function.
5.2.3 Closure model
As mentioned in chapter 4.5.4, macroscopically characterising flow resistance in
porous media, including inertial effects, proposes Forchheimer’s equation, see
Eq. (21). The constitutional variables of the momentum equation are calculated
in the FRV of the system and are volume- averaged over the pore-scale.
〈 〉
The intrinsic quantity of the variable is related to the superficial quantity
through the porosity of the FRV.
〈 〉
〈 〉
COMSOL Multiphysics documentation
A momentum flow balance over the FRV is applied based on the volumeaveraged quantities. The balance considers the applied pressure force on the
boundaries and momentum transport across the boundaries. Shear stresses at
the macroscopic level are generally disregarded at the boundaries of the
computational domain. This was depicted in Eq. (4) and Fig. (8), in chapter 4.1.
The reaction force is volume-averaged over the pore space of the FRV, see Eq.
〈 〉
The term is related to the volume-averaging velocity over the pore space of the
FRV through Forchheimer’s equation, according to Eq. (49).
〈 〉
〈 〉
〈 √
The second-order tensor elements are determined through fitting numerical
data with the least square method. A data set is established by conducting
single-phase simulations in the FRV for various Re-numbers. The interval of
Re-levels included in the data set considers all flow regimes encountered in the
system. The volume force terms ( ) are established based on the Forchheimer
coefficients including viscous and inertial flow resistance in the flow regime.
The resistance coefficients are viewed as 3x3 matrices, if the porous medium
has an anisotropic structure, see Eq. (50). The order of these tensors could be
reduced to zero if the solid matrix of the porous medium resembles an
isotropic arrangement. Moreover, the number of unknown coefficients could
be reduced based on the notion of the isotropic behaviour of the fluid, which
states that the non-diagonal elements with the same yet reversed index in Eq.
(51) are identical.
[ ]
][ ]
][ ]
The left index of and represents the normal direction of the plane in which
the vector is located. The right index represents the direction of the vector
quantity. Hence, the direction and the magnitude of the velocity field will
determine which resistance coefficients are active in the control volume.
The least square method was used to approximate the resistance coefficients in
which the residual is defined as the difference between the actual value, which
is the numerical data of the volume force, and a predicted value described with
the Forchheimer expression. The optimum fitting is achieved when the
summation of the squared residuals are at a minimum. A system of equations is
attained based on using the least square method in Eq. (51).
][ ]
The equation system is presented for a case in which the velocity and the force
term are acting in one direction. As a result of a disordered porous medium in
which velocity is acting in all directions, Eq. (51) extends as the number of
unknowns in the equation system increase from 1 to 9 for each resistance
coefficient in the Forchheimer equation.
In Eq. (52), the viscous and inertial resistance coefficients have been estimated
in relation to the force component in the x-direction. The same procedure has
been executed on the remaining force components. Moreover, the number of
unknown coefficients could be reduced based on the notion of the isotropic
behaviour of the fluid. This would reduce the total amount of unknowns from
18 to 12 coefficients.
5.2.4 Sensitivity studies
The sensitivity of the pore-scale simulations has to be considered based on
three factors, which are the size of the sample domain, the generated mesh grid
and the sensitivity of the resistance coefficients based on amount of numerical
sample points used.
Choosing an appropriate sample volume is essential in volume-averaging
processes of porous media, as the objective is to create volume-averaged
quantities that are independent of the local heterogeneities [82]. The Brinkman
screening length criterion is used to analyse the sample size, see Eq. (53).
The criterion is determined based on the square root of the permeability and is
associated with stokes flow dynamics. In order to strengthen the choice of box
size, symmetry conditions of the original sample volume are extended to the
outer boundaries of a larger sample, see Fig. (20).
The flow characteristics are determined in the FRV by direct numerical
simulations of the NS equations. The resolution of the boundary layer is
therefore critical which demands a sensitivity study of the mesh grid. It is
desirable to establish a model independent of further mesh refinement. It
indicates that for the current boundary conditions, velocity gradients and
pressure of the fluid are calculated with a higher accuracy. An unstructured
mesh grid containing tetrahedral elements was created for the 3-dimensional
model presented, according to Fig. (20).
Fig. (20). Mesh configuration of a disordered fibrous sample domain.
The grid is solved in a finite element space where a set of basis functions are
used to create piecewise linear relations between the mesh elements and
convert them to weak formulations. The mesh parameters that were tested are
the maximum mesh element sizes, the minimum element sizes, the resolution
of narrow regions and the element growth rate, see Table (1).
Table (1). Grid element parameters tested of a 3-dimensional disordered fibrous structure
size [µm]
growth rate
Resolution of
narrow regions
Extra coarse
size [µm]
Coarser (standard)
The mass flux of the porous structure were compared for all of the mesh cases
in order to assess the sensitivity of the simulation at the pore-scale level.
As was stated, the Forchheimer resistance coefficients are estimated from a
numerical data set which includes flow regimes at various Re-levels. The
sensitivity study is conducted by varying the amount of numerical data points
used to approximate these coefficients. The study provides an insight of the
dependency of the resistance coefficients based on estimating over a certain
range of Re-levels.
5.2.5 Volume forces in structureless models
In the up-scaled structureless model, the volume forces in terms of the
Forchheimer coefficients presented in the previous chapter are implemented as
sink terms in the momentum equation, depicted as in Eq. (43). The sink term
is expressed component wise which is presented in Eqs. (54) and (55) for a 2dimensional space.
These expressions are coupled to the local velocity field of the porous medium.
A negative sign is employed on each Forchheimer expression in order for the
force component to act in the opposite direction of the local velocity field, see
Fig. (21).
Fig. (21). Schematic description of the sink term in a 2-dimensional domain in which the
magnitude and the direction of the volume force terms are related to the product of the
local velocity and the flow resistance coefficients.
The ratio of the inlet/outlet surface area and the volume flow rate changes due
to the removal of the internal structure. The constraint in the volume force
model is that the relation between the volume flow rate and the pressure force
is equal to the relation in the FRV-model. Hence, the resistance coefficients are
scaled with the porosity of the medium as a result of the local velocity field
occurring in a structureless medium. The viscous contribution is scaled with the
porosity and the inertial contribution is scaled with the second power of the
porosity, due to the product of two velocity components.
With the elimination of the no-slip condition, the advective and viscous stress
terms in the NS equations are eliminated and replaced with the expression from
Eq. (49).
〈 〉
〈 〉〈√
Eq. (56) is similar to the momentum transport equation presented in Horvat
and Catton [22,23], see Eq. (57).
They state that the pressure force is balanced by shear forces and an empirical
correlation of a local drag coefficient, which represents inertial force as, in their
case, flow resistance across the tubes. The volume force expression in Eq. (56)
constitutes these forces, as we stated that and represent viscous and inertial
forces, respectively.
In the case of implementing volume forces in a two-phase flow system, spatial
variation of fluid properties needs to be considered, due to the existence of two
fluids in the same system. The resistance coefficients in these equations vary,
depending on the type of the fluid. Eq. (58) is therefore implemented in
COMSOL to regulate the coefficients at each phase over the interface
Here is the LS coefficient, which represents the interface thickness and varies
from zero to one, see Eq. (39).
As the internal solid matrix is removed in the volume force model representing
the two-phase flow, capillary forces are excluded. Depending on the
heterogeneities of the pore size, local capillary forces could dominate the
displacement of fluids. This is illustrated in Fig. (22).
Fig. (22). Contour plot of the air/water interface during a dewatering process represented
by a pore-scale (a) and volume force (b) model.
The purpose of the volume force model is to achieve accurate relation between
bulk properties such as flow rate to the system pressure drop. Removal of the
internal structure of the system will vastly reduce the computational time as the
resolution of boundary layers are not the objective of such simulations.
However, any conclusions of local discrepancies of the flow process should
carefully be discussed. The flow rate and pressure drop relation for volume
force models representing a heat exchanger and paper sheet are compared to
their respective FRV-models. This comparison will validate if the
transformation of pore-scale to porous simulations are possible. Moreover, the
computational times for these models are compared in order to assess the
benefit of such models.
There are cases in which the volume force model could be implemented in
systems regarded as porous media, however, with larger length scales. The
contribution of the volume force model is the fact that the computational
domain can be extended, as a result of the reduced computational effect, to
solve the subdomains with implemented volume forces.
5.3 Characterisation of flow resistance in various fibrous structures
This chapter presents the procedure of: generating extensive numerical data for,
in this case, 3-dimensional fibrous media; characterising the Forchheimer based
flow resistance coefficients as a non-dimensional friction factor; validating the
friction factor to classical empirical formulations.
5.3.1 Algorithm for production runs in COMSOL
In the case of repeating simulations with similar build-ups, an approach to
achieve a more efficient overall process is used to automatise the simulation
process from defining the sample volume to exportation of numerical data to a
post processing program, see Fig. (23).
Number of sample
Control of successful CAD and
Fiber sampling
Invert internal
structure and mesh
Reset parameters if error occurs
in CAD or mesh sequence
Initial simulation of the
computational domain
Implementation of numerical
diffusion if convergence issues arise
End session when minimal
diffusion is reached
Setup for simulation of
several Re-levels
Execute simulation of current
Implementation of numerical
diffusion if convergence issues arise
Calculate volume- and surface
integrals for current Re-level
End session when all Relevels are simulated
Export numerical data as text
file for further post processing
End procedure
Fig. (23). Main procedure of the algorithm.
These procedures are executed sequentially in a MATLAB algorithm coupled to
the commercial CFD code COMSOL Multiphysics. In the process of
characterising flow resistance numerically in various fibrous structures, an
algorithm is built up by four general sections. The first section loops a sequence
in which the structure of the domain, boundary conditions and mesh grid
generation are applied until each step is executed successfully. The second
section initiates the simulation process at the lowest Re-level. Implementation
of numerical diffusion is activated if convergence issues arise. The third section
is a setup of a loop, executing several simulations at various Re-levels. For each
Re-level, volume and surface integrals of the computational domain are
calculated. After all Re-level simulations are executed, the numerical data set is
exported as text files for further post processing.
In the fiber sampling blocks in the first section in Fig. (23), an algorithm was
used to generate and uniformly distribute pseudorandom integers for which the
position and orientation for the fibers are determined in the sample volume.
The spatial coordinates and orientation of the fibers are based on three
uniformly distributed variables, respectively, which determine the position of
each fiber and the level of orientation along their respective axis. The algorithm
is able to determine the anisotropic nature of the fibrous structure based on a
few settings, see Fig. (24).
Fig. (24). Various fibrous arrangement in a 3-dimensional sample volume in which (a)
presents an aligned arrangement, (b) presents a layered arrangement in the x-y plane, (c)
presents an isotropic arrangement.
The Brinkman screening length criterion is used to analyse the sample size, see
Eq. (53). According to Clague and Phillips [83], each side of the sample volume
should exceed Brinkman’s screening length criterion times 14 in order to
smooth out local heterogeneities. The length scale is equivalent to the square
root of the permeability. This criterion is vastly used in permeability studies in
3-dimensional fibrous structures [84-86]. The permeability is calculated based on
Stokes’ flow simulations on a considered sample volume. Based on the fiber
diameter and a porosity of approximately 0.7, a sample volume with the
dimension 400x400x400 µm is large enough to meet the Brinkman criterion.
5.3.2 Assessing flow resistance based on Forchheimer coefficients
The approach chosen to quantify flow resistance in these studies is based on
the work of Ahmed and Sunada [87], in which they predict a pressure drop and
flow rate relation with empirical and quasi-empirical correlations. In Macdonald
et al. [78], they state that the porous medium is complex to the extent that the
Forchheimer coefficients ought to be functions of the porous medium, rather
than universal constants. The Forchheimer coefficients are estimated in FRV’s
representing the structure of the porous medium, and hence, related
numerically to the porous medium. A dimensionless friction factor is utilised to
assess flow resistance in these porous mediums. The friction factor is related to
the Forchheimer coefficients through a modified Reynolds expression, see Eq.
(59). The friction factor is deduced by expressing the Forchheimer equation in a
non-dimensional form which presents an expression in correlation with a
modified Re-number.
This equation has been used extensively to correlate experimental data for
various flow conditions and porous materials [77]. Based on this expression, the
modified Re-number is related to the viscous and inertial term of the
Forchheimer equation
These coefficients are not universal, as they have shown to vary with changes in
the porous structure and flow process. The inertial term could be attained by
curve fitting data sets composed by numerical experiments. In that sense,
choosing an appropriate FRV for volume-averaging of velocity and pressure for
various Re-numbers is essential in determining the viscous and inertial term.
Many papers have contributed to establishing valid empirical correlations of the
friction factor and particle Re-number for various cases of packed beds [78,80,8897]. A few expressions are used to compare the numerical simulations conducted
in this work.
) (Montillet)
( )
(a, b and c are fitting parameters)
These empirical correlations are used to compare the friction factor with Eq.
(59) for an aligned disordered distribution of cylindrical fibers in a 3dimensional sample domain.
Results and Discussion
The focus of this chapter is to assess the proposed methodology in establishing
volume forces as sink terms in the momentum equation in order to conduct
time efficient simulations in models regarded as porous media. These bullet
points are the foundation of this chapter:
 Outcome of the fitting procedure.
 Comparison of FRV and volume force models in terms of relating
pressure and flow rates.
 Sensitivity study of box size and mesh configurations.
 Experimental validation of simulation models.
 Assessing flow resistance by comparing numerical data with empirical
formulations and other simulation data.
 Assessing the benefits of implementing time efficient simulations in a
numerical based design process.
6.1 Fitting procedure of resistance coefficients: Paper II, III, IV, V
The approximated resistance coefficients concluding the Forchheimer equation
should be validated in order to justify the fitting procedure. The numerical data
set is used to compare the volume-averaged force term to the force term
constituted by the Forchheimer coefficients. Fig. (25) represents the fitting
result of the force term established for a pin-fin heat exchanger with a R2-value
of 0.999.
Fig. (25). Fitting of the numerical data set to the Forchheimer expression in Eq. (49) for the
pin-fin heat exchanger. The R -value is approximately 0.999.
The comparison made in Fig. (25) is for an ordered structure in which open
boundaries exist in one direction. Hence, the force term acting in that direction
is equal to the balance of momentum, shear and pressure forces in the direction
of the open boundary. In a disordered structure, see Fig. (20), there are open
boundaries in all directions in space resulting in the activation of three force
Fig. (26). Fitting of the numerical data set (blue circles) to the Forchheimer expression
(green dashed line) in Eq. (49) for the paper sheet. The R -value is 0.999, 0.899 and 0.997
for the x, y- and z-direction respectively.
The fitting procedure is successful as presented in Fig. (26). The R2-value based
on comparison of Forchheimer expression of the volume force terms to
numerical data is 0.999, 0.899 and 0.997 for the x, y- and z-direction for the
paper sheet.
6.1.1 Concluding remarks
Overall, the fitting procedure was successfully executed based on the low values
of the estimated error. Moreover, further analysis indicates that increasing
numerical sample points reduces the estimated error.
6.2 Implementation of volume forces as sink terms: Paper II, III, IV
The next step of analysing the validity of the Forchheimer based volume force
terms is implementing them in a structureless domain in the CFD code
COMSOL MultiPhysics. The coefficients are coupled to the local velocity field
according to Eqs. (54) and (55), in which a sink term is established in the
momentum equation. These volume force models are calculated in the same
pressure drop range as their respective FRV-models. In the case of analysing
the heat exchanger, two FRV-models are compared in which FRV1 considers a
small segment of the heat exchanger with periodic condition at the open
boundaries, Fig. (10a). FRV2 considers the actual streamwise length of the heat
exchanger. The comparisons of these models are presented in Fig. (27).
Fig. (27). Comparison of flow rate and pressure drop relation for a structureless model with
volume forces to its respective FRV-model. The mean deviation for the FRV and VF
relations is approximately 4% and 1% respectively.
The volume force models representing the heat exchanger presents a fair
agreement over the range of pressure levels, as the mean deviation is roughly
4% for FRV1 - VF1, and 1% for FRV2 - VF2. Table (2) presents the respective
Re-number for the FRV-model.
Table (2). Variation of flow rate and Re-number for a selected set of simulations.
VVX (air)
∆p [Pa]
̇ [m3/h]
Re [-]
The volume force model representing the paper sheet is exposed to higher
vacuum pressure levels in which the estimation of the Forchheimer coefficients
are calculated for a broader range of Re-numbers. Therefore, the relation of the
mass flux to the pressure levels exhibits a nonlinear relationship compared to
Fig. (27).
Fig. (28). Comparison of mass flux and pressure drop relation for a structureless model
with volume forces to its respective FRV-model. The mean deviation for this case is
approximately 2.9%.
The volume force model presents a fair agreement over the range of pressure
levels as the mean deviation is roughly 2.9%. The Re-numbers for the spectrum
of vacuum pressure levels are presented in Table (3).
Table (3). Variation of water mass flux of and Re-number for a selected set of simulations.
Paper (water)
∆p [kPa]
̇ [kg/m2s]
Re [-]
The deviation between the FRV and volume force model is more pronounced
at higher vacuum levels. This deviation could indicate that a second order
fitting procure of the coefficients is not suitable at the Re-levels presented in
Table (2). Overall, sink terms in the volume force models provides a good
In the case of implementing volume forces in a two-phase flow model, which in
this case is a vacuum dewatering process, the influence of capillary forces are
excluded, as presented in Fig. (22). The vacuum dewatering process was
modelled with the LS-method based on a 2-dimensional pore-scale simulation
with volume forces representing flow resistance in the forming wire, see Fig.
(29a). The other approach is a 2-dimensional volume force model, see Fig.
(29bc) with sink terms representing flow resistance in a 3-dimensional fibrous
space representing the paper sheet, see Fig. (24b). Moreover, the concentration
of fibers packed at the top section of the forming wire is described by
implementing volume forces based on the same Forchheimer coefficients in the
lower section of the paper sheet, see Fig. (29c). These sink terms are, however,
regulated with a time constant which states that there are no volume forces
present at the start of the process, and they gradually increase to a maximum
value which represents a final packing state. A simplified representation of the
forming wire is presented in Fig. (29c) in order to analyse the influence of
physically reducing the open area. These models simulate dewatering of a sheet
with a basis weight of 50 g/m2 and all models has a porosity of approximately
0.72 and a basis weight at 50 g/m2.
Fig. (29). Representation of a paper sheet with a basis weight at 50 g/m . Three models
are presented in which (a) a pore-scale model with volume forces is representing flow
resistance in the wire. Model (b) presents flow resistance in the paper sheet and the wire
with volume forces. Model (c) presents flow resistance in the paper sheet with volume
forces and flow resistance representing fiber packing at the top layer of a simplified wire
Fig. (30). Dry content and dwell time relations for three simulation models (Fig. (29)) of a
paper sheet with basis weight 50 g/m .
Based on the cases presented in Fig. (30), it seems that implementing volume
forces at the top region of the simplified wire representation (red dashed line)
presents the highest flow resistance as the dry content is not increasing at the
same rate. Some interesting remarks on this distinction are that volume forces
in model (a) and (b) in Fig. (29) seems to affect the dewatering process slightly.
In model (a), volume forces are determined based on a trivial representation of
the forming wire in 3-dimensions. Numerical results from models in Fig. (29ab)
indicate that flow resistance due to the wire is neglectable indicating that
averaging over a coarse solid arrangement is pointless in case of denser regions
in the system controlling the permeability. In model (b), neither volume forces
representing the paper sheet and the wire seems to influence the dewatering. In
model (c) however, the system reacts on reducing the open area which forces
the fluid to spatially accelerate which increases the volume forces according to
Eqs. (53) and (54).
The influence of excluding capillary forces is visibly illustrated by analysing the
dewatering rate in relation to the dwell time, see Fig. (31). The models in
comparison are from Fig. (29ac). The first distinction made between the models
is that the pore-scale model exhibits various peaks of the dewatering rate
whereas the volume force model presents a single peak at the start of the
process. The first peak in the pore-scale model represents the time for air
penetration as the dewatering radically decreases afterwards. The other
following peaks present infrequent water release from dense regions which
capillary forces dictate. The peak in the volume force model represents the
effect of fibers packing at the top layer of the forming wire. The volume force
model presents a smooth dewatering rate.
Fig. (31). Comparison of the dewatering rate between the pore-scale and volume force
model. These models presents dewatering in a paper sheet with a basis weight at 50 g/m .
The computational time of the volume force models is significantly shortened
as the geometrical complexity of the simulation domain is reduced, along with
the amount of DOF. Table (4) presents a list in which the computational time
of simulation models based on volume forces as sink terms are compared to
pore-scale (FRV) models of the same system. The comparison is made for both
single and two-phase flow processes, steady state and transient systems, as well
as 2 and 3-dimensional models.
Table (4). List of computational time for pore-scale and volume force models
CPU time
Heat exchanger
58.82 min
Paper single-phase 13.38 min
Paper two-phase
846.0 min
Volume force model
CPU time
0.25 min
0.53 min
23.0 min
6.2.1 Concluding remarks
In single-phase simulations, volume forces are successfully implemented based
on achieving acceptable flow rate and pressure drop relations in comparison to
their respective FRV-model. In two-phase flow simulations, capillary forces are
excluded along with the removal of the internal structure which affects the
dynamics of the dewatering process. The dry content and dwell time relations
could still be valid in terms of predicating the bulk properties of the system.
However, the details of the dynamics are eliminated in the process.
The computational time for the volume force based models were reduced
significantly, as presented in Table (4). This provides time efficient simulations
during pressure drop and volume flow rate predictions in the case of the heat
exchanger and prediction of dry content and dwell time relation based on
system parameters in a vacuum dewatering process. Certainly, it should be
possible to predict such relations in up-scaled systems with the same porous
configurations. Another concluding remark on time efficient simulation is the
implementation of the volume force model in larger computational domains as
a result of reducing DOF of the entire system. For instance, the internal duct
system presented in Fig. (3).
As it was stated before, the details of the flow process is eliminated which
should be considered when implementing a volume force model in a lager
system. The reason is that subsystems following a volume force model could be
miscalculated due to the homogenous flow patterns entering the system. For
instance, dealing with a system with high Re-numbers, turbulence is an essential
parameter. As a result of smoothing out heterogeneities, the build-up of
turbulence is mistreated which affects key inlet parameters such as turbulence
intensity and temperature distribution.
6.3 Sensitivity study of pore-scale simulations: Paper II, III, IV, V
Several aspects of conducted sensitivity studies should be considered in a
modelling process heavily based on numerical analysis of constitutional laws of
physics. Besides experimental validation, which is a requirement in numerical
analysis, sensitivity studies of simulations models and processes should be
analysed further.
The sensitivity of the mesh grid is presented as the estimated Forchheimer
coefficients in the flow directions are presented for cases in Table (1). The
variation of the mass flux is presented in Fig. (32). The mass flux is defined
based on the volume-averaged flow velocity.
Fig. (32). Mass flux and vacuum level relation for various mesh grid configurations.
The mass flux based on the volume-averaged z-component velocity increases as
the grid is more refined. At the highest vacuum levels, the mass flux increases
with approximately 19%, as the grid is refined to the settings labelled as
“coarser”. The next refining steps yield 8% and 5%, respectively.
The resistance coefficients acting in the flow direction are chosen as parameters
for validation; see the coefficients with a double index of z-components in Eq.
(50). At low sample points, the coefficients fluctuate considerably and
unpredictably, indicating that the set of data points is sparse. As the amount of
sample points are increased, the value of the resistance terms seems to stretch
towards a certain value with no fluctuations along the increasing sample points,
see Fig. (33).
Fig. (33). Variation of estimated viscous and inertial resistance coefficients based on the
amount of numerical sample points. The maximum Re-number increases with the amount
of sample points.
The friction factor is based on the Forchheimer coefficients, which are
averaged over a set of data points. The modified friction factor and Re-number
relation are examined for a range of sample points which is presented in Fig.
(34). It is obvious that the friction factor is depending on the range and
frequency of Re-numbers indicating that it is sensitive to the amount of sample
points. The factor is converging towards a fixed value as the deviation of the
curves in Fig. (34) in decreases along with the higher sample points. The
sensitivity is visible at low Re-numbers as the friction factor for all sampling
cases seems to increase towards the same value.
Fig. (34). Modified friction factor and Re-number relation based on the amount of sample
The sample points presented in Figs. (33) and (34) are equivalent to Renumbers ranging from 0.002 to 1000.
6.3.1 Concluding remarks
A few concluding remarks are that the sensitivity analysis of the box size and
mesh grid should be conducted early in order to assess the robustness of the
representative structure as well as the resolution of the flow field. Increasing the
box size and grid resolution would yield a more robust FRV-model, as well as
increase computation time. Therefore, refinements should be made wherein
both these requirements are met.
It is obvious that the friction factor is sensitive to the amount of numerical
sample points used in the estimation of the Forchheimer coefficients. As
sample points are increasing, the Forchheimer coefficients stabilises towards a
constant value. The variation is still hard to predict as analysis has indicated that
the Forchheimer coefficients are dependent on the porosity and level of
orientation in the fibrous structure.
6.4 Experimental validation of simulation models: Paper II, III, IV
Volume forces presented as sink terms in the NS equations were implemented
in the heat exchanger unit in the absence of the internal pin-fin configuration.
A comparison of the magnitude of the volume flow rate and pressure drop is
presented in Fig. (35).
Fig. (35). A comparison of the pressure drop and volume flow rate relation of the volume
force models to experimental data.
The volume force model based on FRV1 significantly deviates from the
experimental data due to the absence of inlet regions of each heat exchanger,
which in turn excludes the high viscous shear stresses as the boundary layers are
developed along the fins.
The volume force model presented in Fig. (29c) is further evaluated by
comparing the dewatering rate with experimental data from Rezk et al. [98] and
experimental data from Pujara et al. [44].
Fig. (36). Dewatering rate and dwell time relations at three different vacuum levels for the
volume force model to experimental data from Rezk et al. (2013) and experimental data
from Pujara et al. (2008b). Comparison is made for paper sheet with a basis weight of 50
g/m .
The dynamics of the dewatering rate in the experimental setups are still hard to
comprehend, due to the sample frequency. However, experimental data
indicates that the rate of packing at the contact surface of the wire plays an
essential role in the process, as the dewatering rate decreases with the dwell
time. The sudden dip of the dewatering rate for the volume force models in
Fig. (36) is a result of air penetrating the paper sheet. The models indicate that
reducing the vacuum level delays the time for air penetration. There is a slight
offset between the first sample points and the peaks of the curves which could
be a result of the time delay of sampling. The offset is visible, due to the fact of
the short time span of the dewatering process.
6.4.1 Concluding remarks
Comparison to experimental data indicates that the proposed volume force
models predict the flow process convincingly. (i) The volume force model for
the heat exchanger unit based on averaging over the FRV2-model shows a fair
agreement, as the flow rate and pressure drop relation differed by
approximately 8% from the experimental data. (ii) The simulated values for the
dewatering rate as a function of time showed a fair agreement with the
experimental data, as the data shows a rapid decreasing dewatering rate.
6.5 Assessment of flow resistance coefficients: Paper V
The aim with the validation in this chapter is to assess the flexibility of the
algorithm, in which volume forces are characterised based on the averaging
procedure presented in chapter 5.2.
In the algorithm presented in Fig. (23), the simulated flow regimes can be predetermined by configuring the Re-number with parameters such as material
properties of the fluid, porosity and pressure loss over the simple domain.
Depending on the system at hand, the algorithm could be configured to create
numerical data sets at highly turbulent flow regimes, as well as flow regimes
with Re-numbers well below creeping flow conditions. In this thesis, the
friction factor is investigated for aligned fibrous structure with porosities at 0.7
to 0.8.
In Fig. (37), the friction factor is estimated with Re-numbers ranging from
approximately 0.0005 to 1000 for an aligned disordered fibrous arrangement
with a porosity at 0.8, see Fig. (24a). The figure displays the numerically
determined flow resistance coefficients over a wide range of flow regimes in
comparison to the formulations of Ergun [80], Montillet et al. [90] and Tamayol et
al. [96], which consider both viscous and inertial effects. The expressions from
Ergun [80], Montillet et al. [90] are based on the 3-dimensional packing of spheres.
Montillet et al. [90] concluded their empirical formulations based on packed beds
of spheres also accounting for the bed to particle ratio and level of packing,
which is determined by the bed porosity. Tamayol et al. [96] established their
formulation based on numerical simulations, see Eq. (64). These simulations are
based on a 3-dimensional ordered fibrous structure representing a simple cubic
arrangement. Based on Fig. (37), the presented friction factor is in fair
agreement with Erguns formulation.
Fig. (37). Logarithmic plot of the friction factor and Re-relation for a wide range of Renumbers, ranging from 0.0005 to1000 in an aligned fibrous structure with a porosity at 0.8.
A comparison of the friction factor and Re-number relation between the numerical
simulations and Blake-Kozeny expression shows the departure from the linear relation
between the friction factor and Re-number.
In Fig. (37), the numerical friction factor is in good agreement with the BlakeKozeny (BK) expression, which solely considers viscous dissipation and is
expressed for packed bed columns as
The expression is valid for Re < 10, which is visible in Fig. (37), as the
presented friction factor along with the expression from Ergun [80] are visibly
deviating as the Re-number reaches the value 10. Fig. (38) presents the friction
factor for an aligned structure at a porosity of 0.7.
Fig. (38). Logarithmic plot of the friction factor and Re-relation for a wide range of Re-6
numbers ranging from approximately 10 to 1000 in an aligned fibrous structure with a
porosity at 0.7.
The figure illustrates a better comparison between the presented friction factor
and the expression presented by Tamayol et al. [96]. Fair comparison with
Erguns formulation was observed for both porosities. However, the expression
presented by Montillet et al. [90] underestimates the friction factor in both cases.
By analysing the sensitivity of the sample points presented in chapter 6.3, it is
obvious that comparisons to classical empirical correlations is not generalised
for a number of cases regarding the porosity.
Current numerical data from Yazdchi and Luding [59] was compared to our
simulations in Fig. (39). Yazdchi and Luding analysed flow resistance for 2dimensional disordered unidirectional fibers in which their numerical data
presents friction factors at porosities of 0.7 and 0.8.
Fig. (39). Logarithmic plot of the modified friction factor and Reynolds number relation in
comparison with numerical data from Yazdchi at porosities of 0.7 and 0.8. Comparisons
are made for aligned disordered fibers.
The factor concerning the porosity at 0.7 is overestimated, compared to the
simulations of Yazdchi and Luding. They estimated their friction factor at Renumbers ranging up 30; whereas our friction factors are fitted for a broader
range which results in an offset. However, reducing our data set at the high Renumbers resulted in a better comparison, which is explained in Fig. (34).
Overall, the successful comparison is satisfactory and the observation
strengthens the flexibility of the algorithm presented in Fig. (23), which could,
in a time efficient manner, generate extensive numerical data sets and accurately
estimate flow resistance coefficients for further usage in time efficient
simulations of flow in porous media for single and two-phase flow, as well as
stationary and time dependant flow processes.
6.5.1 Concluding remarks
A few concluding remarks are: (i) The comparison to the classical empirical
correlation by Ergun display a fair agreement to the presented friction factor.
However, empirical formulations seem to deviate from the presented friction
factor depending on the range of Re-numbers the resistance coefficients are
fitted over and the porosity of the fibrous structure. (ii) The friction factor for
aligned fibrous structures is in good comparison with the numerical study of
Yazdchi and Luding provided that the estimation of the friction factor is
determined over the same range of Re-numbers.
6.6 Assessment of numerical based design process: Paper I
In commercial CFD codes a single user can perform different tasks, such as
designing geometry (CAD) and simulating a physical phenomenon at various
degrees of complexity. However, when dealing with the design of industrial
components and systems, the geometry becomes complex and in many cases
extensive. It is reasonable to believe that working with such a geometry design
based on fluid mechanic parameters requires knowledge in different fields.
Depending on the material and the magnitude of the project, researchers and
engineers of different fields are usually involved. In large-scale companies, it
may be more effective to create a virtual experiment system that consists of the
integration of various elements of technology, something which is
acknowledged by Koikekoi [1]. Small-scale projects may, however, be utilised
When working with a heuristically based design process, several geometric
modifications are conducted based on guesses. Without experimental validation
or indicative parameters, modifications such as these cannot be assessed
properly. When working with CFD, it is always beneficial to use an
experimental laboratory for validation. Then again, certain components are
intricate to measure. The establishment of design criteria is, together with visual
analysis, used to assess modifications. It should be noted, however, that the
validity of the simulation in terms of absolute values such as pressure drop is
questionable, due to certain assumptions on boundary conditions and
approximations on numerical instabilities. In a design process, such as the one
described in Paper I, the issue is to make improvements based on previous
models. A constant process of knowledge sharing is established, as there is a
continuous verbal communication about the dependency of geometry structure
and flow patterns. A summary of the beneficial aspects of implementing CFD
in a design cycle are presented in Fig. (40).
Design cycle
Heat pump
tumble dryer
Paper machine
Internal duct
Spatial resolution
of flow field
Few measurment
Extensive data set
of flow process
Few sample
Trial and error
analysis of system
Trial and error
Few prototype
Time efficient
Time and
demanding exp.
Fig. (40). Flow chart describing the impact of implementing CFD analysis in the application
fields encountered in this thesis.
The significance of the knowledge sharing is noticeable, as small-sized industrial
companies generally lack a fluid mechanic knowledgebase. As a design process
is completed, the mechanical engineer passes on the knowledge to further
design projects. It is hence, a favourable approach to sustainable development.
In addition, it is beneficial for a company to work with this design process. It is
time consuming to design several prototypes and conduct flow and pressure
measurements. In addition, it generally lacks the detailed analysis that is
important to create a relation of fluid and geometric parameters. It is therefore,
profitable to take a numerical based approach.
The benefits of implementing CFD analysis on the vacuum dewatering system
are similar to those presented for the heat pump tumble dryer. This includes the
generation of data sets presenting a highly detailed resolution of the dynamics
in the flow process. Along with establishing time efficient simulation models, a
thorough analysis of system parameters is possible, which provides an insight in
the process which a classical experimental analysis would fail to do. However, it
should be pointed out that experiments are crucial in these design cycles in
terms of validating simulation data.
The involvement of CFD, however, requires analysis of the resolution and the
sensitivity of the mesh grid. In more or less every CFD study, a sensitivity
analysis of the mesh grid is required, as it plays an essential role in presenting
fluid mechanic parameters. There are no specific guidelines for creating certain
mesh grids, as flow processes take various forms depending on the geometry of
the domain. Hence, establishing a good feel for meshing requires experience in
modelling. Moreover, the demand for computer power is reduced by doing so.
6.6.1 Concluding remarks
A few concluding remarks based on the assessment made in previous chapter is
that; (i) Implementing CFD-models in a design cycle generally yields extensive
data points with high resolution in space and time compared to experiments; (ii)
Thorough analysis can be conducted on the system instead of the traditional
trial and error approach. Moreover, it is possible to share knowledge based on
this approach. Time efficient simulations can reduce manufacturing of
prototypes and extensive experimental runs, both of which are time and
resource consuming factors.
The emphasis of the research conducted in this thesis is to present methods for
reducing the complexity of geometrical structures. The focus is to establish time
efficient simulations in which the concept is to reduce the geometrical
complexity of systems by replacing the internal structure with volume forces as
sink terms in the momentum equation.
The flow rate and pressure drop relations are well predicted in the volume force
models to their respective FRV-simulations. The relation for the heat exchanger
presented a mean deviation of approximately 4% whereas the relation for the
paper sheet presented a mean deviation of approximately 2.9%. The reliability
of these comparisons are based on establishing FRV-models with a consistent
mesh grid configuration and a sufficient amount of numerical data points for
volume force characterisation.
The method of establishing volume forces as sink terms are strengthened based
on the comparison to experimental data in which the pressure drop and the
dewatering rate are fairly predicated in the heat exchanger and paper sheet
respectively. The comparisons indicate that the bulk properties of the system
are well predicted, however, the implementation of volume forces eliminates
the spatial resolution of the detailed system. In two-phase flow cases, capillary
forces are excluded along with the internal solid matrix. Moreover,
considerations have to be made when implementing a volume force model as a
subdomain in a larger control volume. Key inlet parameters such as turbulence
could be mistreated and have significant effects on the following domain.
The comparison to classical empirical correlations seem to deviate from the
presented friction factor depending on the range of Re-numbers the resistance
coefficients are fitted over and the porosity of the fibrous structure. In this
thesis however, the formulation by Ergun displays a fair agreement to the
presented friction factor for an aligned fibrous structure. Moreover, the friction
factor for aligned fibrous structures is in good comparison with the numerical
study of Yazdchi and Luding provided that the estimation of the friction factor
is determined over the same range of Re-numbers.
The computational time of the volume force models was significantly shortened
as the geometrical complexity of the simulation domain was reduced, thus
reducing the amount of DOF. The volume force model of the heat exchanger
successfully predicted pressure drop and flow rates at simulation times of 15
seconds, wherein the computational time of the respective FRV-model took
approximately 1 h. Given the fact that the FRV-model only considers a small
segment of the heat exchanger, resolving the entire solid matrix would yield an
unfavourable amount of DOF. The reduction in computational time enabled
time efficient simulations when determining the size of heat exchangers for a
given system, with a demand on a certain flow rate and pressure drop. Even
though there is a wide access of empirical formulation on flow rate and
pressure drop relations for various types of heat exchangers, this method
should be applicable on new design configurations in order to quantify such
The computational time of a two-phase flow model of the paper sheet with a
basis weight of 50 g/m2 was approximately 14 h, wherein implementing volume
forces in the paper model reduced computational time to roughly 23 min. In
turn, key system parameters in a vacuum dewatering process could be analysed
in a more time efficient manner. Such parameters include variation of vacuum
levels and duration of pulses, different basis weights, and various arrangements
of the forming wire. These parameters are analysed experimentally these days in
pilot-scale machines which require preparation, extensive measurement
equipment and facilities, pulp and water among other resources. Reducing
experimental runs based on accurate numerical models would reduce the time
and cost requirements during the design and development cycle. Moreover,
dimensioning the components more accurately of a full-scale paper machine
would reduce the probability of failed production runs which would result in a
considerable amount loss of income.
Future Research
There are several interesting issues that have yet to be studied in this thesis. The
method of implementing volume forces have solely been focused on in the
momentum equation. Naturally, the next step in the analysis of the heat
exchanger unit is to investigate the possibility of solving the energy equation
with a similar approach. Hence, certain options will be examined in replacing
the tube bundles with a heat source and sink. In order to do so, the NS
equations are required to couple with the heat transfer equations. The
establishment of a volume force model coupled with heat transfer enables the
user to analyse the efficiency of the heat exchanger unit using time efficient
numerical simulations.
In the presented results, the interaction between the paper sheet and forming
wire seems to affect the dewatering process the most. Based on this conclusion,
a more physically representable configuration of the forming wire should be
investigated further. Moreover, capillary forces have yet to be included in the
volume force models. Moving forward, these forces should be analysed further
in order to improve the prediction of the models.
Even though successful comparison of the friction factor has been established
to classically empirical relations, the sensitivity of the resistance coefficients in
terms of quantifying them in a certain Re-level require more experimental data
in order to strengthen their validity. Overall, there is still a lack in the literature
on the knowledge of these resistance coefficients at higher Re-numbers;
variation of the coefficients based on the amount and frequency of sample
points at higher Re-numbers; variation of the coefficients with the arrangement
of 3-dimensional disordered fibrous structures. Based on the established
numerical algorithm presented in chapter 5.3.1, an extended set of numerical
experiments can be conducted with increased sample points in order to gain
understanding regarding the issues just stated.
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Methods for Reducing the Complexity
of Geometrical Structures Based on CFD
As there is an ongoing expansion of CFD usage in industry, certain issues need
to be addressed as they are becoming more frequently encountered. The general
demand for the simulation of larger control volumes and more advanced flow
processes result in an extensive requirement of computer resources. Moreover, the
implementation of commercial CFD codes in small-scaled industrial companies
seems to generally be utilised as a black box based on the knowledge of fluid
mechanic theory. Increased partnerships between industry and the academic
world involving various CFD based design processes generally yield to a verbal
communication interface, which is a crucial step in the process given the level of
dependency between both sides.
Based on these notions, a method for establishing time efficient CFD-models
with implementation of volume forces as sink terms in the momentum equation
is presented. The internal structure, or parts of the structure, in the simulation
domain is removed which reduces the geometrical complexity and along with it,
computational demand. These models are the basis of assessing the benefits of
utilizing a numerical based design process in industry in which the CFD code
is used as a communication tool for knowledge sharing with counterparts in
different fields.
ISBN 978-91-7063-565-6
ISSN 1403-8099
DISSERTATION | Karlstad University Studies | 2014:32
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