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Direct numerical simulation of dense gas-solid non-isothermal flowsTavassoli Estahbanati, H.
DOI:10.6100/IR782478
Published: 01/01/2014
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Citation for published version (APA):Tavassoli Estahbanati, H. (2014). Direct numerical simulation of dense gas-solid non-isothermal flowsEindhoven: Technische Universiteit Eindhoven DOI: 10.6100/IR782478
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Direct numerical simulation of dense gas-solid
non-isothermal flows
Direct numerical simulation of dense gas-solid
non-isothermal flows
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de
Technische Universiteit Eindhoven, op gezag van de
rector magnificus, prof.dr.ir. C. J. van Duijn, voor een
commissie aangewezen door het College voor
Promoties in het openbaar te verdedigen
op maandag 1 december 2014 om 14:00 uur
door
Hamid Tavassoli Estahbanati
geboren te Shiraz, Iran
Dit proefschrift is goedgekeurd door de promotoren en de samenstelling van de
promotiecommissie is als volgt:
voorzitter: prof.dr.ir. J.C. Schouten
promotor: prof.dr.ir. J.A.M. Kuipers
copromotor: dr.ir. E.A.J.F. Peters
leden: prof.dr.ir. E.H. van Brummelen
prof.dr. J.G.M. Kuerten
prof.dr.ir. B.J. Boersma (Technische Universiteit Delft)
prof.dr.ir. C.R. Kleijn (Technische Universiteit Delft)
prof.dr. J. Frohlich (Technische Universiteit Dresden)
To my parents
Dit proefschrift is goedgekeurd door de promotoren:
prof.dr.ir. J.A.M. Kuipers
Copromotor:
dr.ir. E.A.J.F. Peters
The research reported in this thesis was funded by the European Research Council,
under its Advanced Investigator Grant scheme, contract number 247298 (Multiscale
Flows).
Copyright c© 2014 by Hamid Tavassoli, Eindhoven, the Netherlands.
No part of this work may be reported in any form by print, photocopy or any other
means without written permission from the author.
Publisher: Gildeprint, Enschede
A catalogue record is available from the Eindhoven University of Technology Library
isbn 978-90-386-3743-3
Table of contents
Table of contents vii
Summary ix
Samenvatting xiii
Nomenclature xvii
1 Introduction 1
1.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Computational Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Multi-scale modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Heat transfer in fluid-particle systems 7
2.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Heat transfer coefficients in a random array of particles . . . . . . . . 8
2.4 Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 Heat transfer correlations . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.6 DNS of fluid-particle heat transfer . . . . . . . . . . . . . . . . . . . . 14
3 The Immersed Boundary Method 19
3.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4 Numerical solution method . . . . . . . . . . . . . . . . . . . . . . . . 26
3.5 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4 DNS of random arrays of monodisperse spheres 37
vii
viii Table of contents
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2 Simulation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.4 Influence of micro-structure on particle-fluid heat transfer rate . . . . 45
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5 DNS of bidisperse spheres 57
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2 Bidisperse systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6 DNS of non-spherical particles 71
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.2 Physical model and numerical method . . . . . . . . . . . . . . . . . . 73
6.3 Heat transfer correlations in packed and fluidized beds . . . . . . . . . 75
6.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7 Summary and recommendations 83
7.1 Summary and general conclusions . . . . . . . . . . . . . . . . . . . . . 84
7.2 Outlook and recommendations . . . . . . . . . . . . . . . . . . . . . . 85
References 87
Acknowledgements 95
Curriculum Vitae 97
Summary
Direct Numerical Simulation of heat transfer in
gas-solid systems
Non-isothermal gas-solid flows are widely used in a variety of industrial applications
such as packed and fluidized bed reactors. In order to arrive at an optimal design and
control of such systems, a precise prediction of the temperature distribution, as well
as the flow field inside the equipment is necessary. Over the past decades, a lot of
studies have focused on the heat transfer in gas-solid flows and many experiments have
been conducted to establish empirical heat transfer coefficients (HTC). Usually these
experiments were conducted under a specific range of operating conditions where the
results are interpreted with the help of simplified models. Although these correlations
have been used successfully for design purposes, these are not generally applicable for
different systems and a wider range of operating conditions. Beside these issues, the
correlations have limitations in providing insight into the complex thermal dynamics.
For example, the contribution of each of the three basic heat transfer mechanisms
(convection from fluid, conduction from particles and radiation) on the total HTC
is difficult to determine. Moreover, the HTC depends strongly on the gas-solid flow
pattern. Also the impact of micro-structural information on the HTC is not clear
and cannot be quantified easily with an empirical heat correlation. In other words,
the empirical correlations only provide a description of the average thermal behavior
of a system.
With the increase of computational power, Computational Fluid Dynamics (CFD)
simulation has become an attractive and popular method for gaining in depth knowl-
edge on transport phenomena in multiphase reactors. The well-known CFD tech-
niques for simulation of particulate flows proposed in the literature can be classified in
three groups: Two-Fluid method (the motion of each phase is governed by a separate
set of Navier-Stokes equations; the interaction between the phases is approximated
by empirical correlations), Discrete Element method (the fluid motion is described
by the Navier-Stokes equations and each particle is described in terms of Lagrangian
equations of motion; the interaction between the phases is represented with a clo-
ix
x Summary
sure model) and Direct Numerical Simulations (the fluid and particulate phases are
treated by considering the Navier-Stokes equation and the Lagrangian equations of
motion, respectively; the interaction between the phases is enforced through the no-
slip boundary condition at the surface of the particle, and hence there is no need for
empirical closures).
With the increase of computational power, Direct Numerical Simulation (DNS)
approaches have been employed to simulate complex systems involving multiphase
flows. In the DNS approach, the fluid and solid phases are treated by considering the
Navier-Stokes equations and the Newtonian equations of motion, respectively. The
mutual interactions between the phases are obtained by enforcing the appropriate
boundary conditions at the surface of the particle (e.g. no-slip and Dirichlet boundary
condition for momentum and heat transfer, respectively) and in principle no empirical
correlations are required. Consequently the DNS approach can improve our insight
regarding the effect of non-ideality on the hydrodynamic and thermal behavior in
multiphase flows. In this study, we employ the Immersed Boundary Method (IBM) to
simulate non-isothermal flows through stationary arrays with random (non-)spherical
particles.
In chapter 4, the IBM method proposed by Uhlmann was extended for Direct Nu-
merical Simulation of non-isothermal fluid flow through dense fluid-particle systems.
A fixed Eulerian grid is employed to solve the momentum and energy equations by
traditional computational fluid dynamics methods. Our numerical method treats the
particulate phase by introducing momentum and heat source terms at the boundary
of the solid particle, which represent the momentum and thermal interactions between
fluid and particle.
Although many studies have been devoted to investigate drag and heat transfer
characteristics in packed and fluidized beds, the associated transfer coefficients have
been obtained mainly for ideal cases (e.g. uniform arrangement of particles, spher-
ical particles, monodisperse particles). Therefore, it would be unrealistic to expect
that such hydrodynamic and thermal models are able to cover all types of non-ideal
industrial systems. In chapter 5 and 6 the heat transfer in random fixed arrays of
bidisperse spheres and sphero cylinders were, respectively, investigated. The objective
of this study is to examine the applicability of well-known heat transfer correlations,
that are proposed for spherical particles, to the systems with polydisperse spherical
or non-spherical particles.
In chapter 5, on the basis of the extensive DNSs, it was found that that the
correlation of the monodisperse HTC can estimate the average HTC of bidisperse
systems if the Reynolds and Nusselt numbers are defined based on the Sauter mean
diameter.
In chapter 6 sphero-cylinders are used to construct the beds. The numerical
results show that the heat transfer correlation of spherical particles can be applied to
xi
all test beds by using a properly chosen effective diameter in the correlations for the
non-spherical particles. Our results show that the diameter of the sphero-cylinder is
the proper effective diameter for characterizing the heat transfer.
We expect that these new results will significantly improve the numerical modeling
of particulate flows using multi-scale approaches.
Samenvatting
Directe Numerieke Simulatie van Warmtetransport
in Gas-Vast Systemen
Niet-isotherme gas-vast stromingen komen veelvuldig voor in industriele toepassingen
zoals gepakt-bed en gefluıdiseerd-bed reactoren. Een goede voorspelling van zowel de
temperatuurverdeling alsook de stroming is nodig voor een optimaal ontwerp en een
goede regeling van dit soort systemen. De afgelopen decennia is er veel onderzoek
gedaan naar warmtetransport in gas-vast stromingen en is er menig experiment uit-
gevoerd om warmteoverdrachtscoefficienten (HTC) te bepalen. Meestal werden dit
soort experimenten verricht voor een beperkte range van bedrijfsvoeringen, en wer-
den de resultaten geınterpreteerd met behulp van gesimplificeerde modellen. Ook al
zijn de bepaalde correlaties succesvol gebruikt voor ontwerp van processen, ze zijn
niet bruikbaar voor een breder scala van systemen en ook niet voor een grote range
aan bedrijfsvoeringen. Een andere kwestie is dat ze weinig inzicht verschaffen in het
complexe thermische gedrag. Het is, bijvoorbeeld, moeilijk te bepalen wat de bij-
dragen van de drie elementaire warmtetransport mechanismen (convectie, geleiding
en straling) aan de totale HTC zijn. Daarnaast is de HTC erg afhankelijk van het
specifieke karakteristieken van de gas-vast stroming. De invloed van de microstruk-
tuur op de HTC is niet duidelijk en kan niet gemakkelijk gekwantificeerd worden aan
de hand van een empirische correlatie. Met andere woorden, empirische correlaties
geven alleen een beschrijving van het thermische gedrag op grofstoffelijk niveau.
Met de toename van de rekenkracht van computers is numerieke stromingsleer
(CFD) een aantrekkelijke en populaire methode geworden om gedetailleerde kennis
te vergaren over transportverschijnselen in meerfase reactoren. De welbekende CFD
technieken voor deeltjes stromingen zoals die in de vakliteratuur besproken worden
zijn: het Twee-Vloeistoffen Model (de beweging van elke fase wordt beschreven door
een tweetal Navier-Stokes vergelijkingen; de interactie tussen de fasen wordt bena-
derd m.b.v. empirische correlaties), het Discrete-Deeltjes Model (het fluıdum wordt
beschreven door een Navier-Stokes vergelijking en elk afzonderlijk deeltje door een
Lagrangiaanse bewegingsvergelijking; de deeltjes botsen en de deeltjes-fluıdum inter-
xiii
xiv Samenvatting
actie m.b.v. empirische correlaties) en Direct Numerieke Simulatie (het fluıdum en
de deeltjes fasen worden beschreven m.b.v., respectievelijk, een Navier-Stokes verge-
lijking en Lagrangiaanse bewegingsvergelijkingen; de interactie tussen de fasen wordt
bepaald door het opleggen van no-slip randvoorwaarden en er is dus geen sluitings-
relatie nodig.)
Met de toename van de rekenkracht is het gebruik van Directe Numerieke Simula-
tie (DNS) van complexe systemen met meerfase stromingen aantrekkelijk geworden.
In de DNS benadering worden de fluıdum en de vaste fasen beschreven m.b.v., resp.,
de Navier-Stokes vergelijking en Newton vergelijkingen. De wederzijdse interactie
tussen de fasen wordt verkregen door het opleggen van randvoorwaarden op de deel-
tjes oppervlakten (bijv. no-slip en Dirichlet randvoorwaarden voor, resp., impuls- en
warmtetransport) en er is, in principe, geen empirische correlatie nodig. Daardoor kan
DNS ons inzicht verbeteren in de gevolgen van niet-idealiteiten op het hydrodynami-
sche en thermische gedrag in meerfase stromingen. In dit onderzoek gebruiken we de
‘Immersed Boundary Method’ (IBM) om niet-isotherm stromingen door ongeordende
collecties van (niet-)bolvormige deeltjes te simuleren.
In hoofdstuk 4, zal de IBM methode van Uhlmann uitgebreid worden naar Di-
recte Numerieke Simulatie voor niet-isotherme stroming in gas-vast systemen. Een
onveranderlijk Euleriaans raster wordt gebruikt om de impuls en energievergelijkingen
op te lossen m.b.v. traditionele numerieke stromingsleer methoden. Onze numerieke
methode implementeert de deeltjesfase door de introductie van impuls en warmte
brontermen op de randen van de deeltjes, die de impuls en thermische interacties
tussen fluıdum en deeltjes bepalen.
Er zijn vele studies die stromingsweerstand en warmtetransport in gepakte en
gefluıdiseerde bedden hebben onderzocht, maar meestal worden warmtetransport
coefficienten voor ideale gevallen bepaald (bijv. netjes geordende, bolvormige deel-
tjes met allemaal dezelfde grootte). Het is daarom onaannemelijk dat deze hydro-
dynamische en thermische modellen alle typen van niet-ideale industriele systemen
kunnen beschrijven. In hoofdstuk 5 en 6 wordt de warmtetransport onderzocht in
niet-geordende verzamelingen van, resp., bi-disperse bollen en sphero-cilinders. Het
doel van deze studie is om de toepasbaarheid te onderzoeken van welbekende warmte-
transport correlaties voor bolvormige deeltjes in het geval dat het systeem polydispers
is of uit niet-bolvormige deeltjes bestaat.
In hoofdstuk 5 wordt, op basis van een groot aantal simulaties, geconcludeerd dat
de ‘monodisperse’ correlatie toepasbaar is voor het berekenen van de gemiddelde HTC
voor bidisperse systemen als het Reynolds en het Nusselt getal worden gebaseerd op
de Sauter gemiddelde diameter. In hoofdstuk 6 worden bedden met sphero-cylinders
gebruikt. De numerieke resultaten laten zien dat de warmtetransport correlaties
van bolvormige deeltjes kunnen worden gebruikt door de juiste keuze te maken voor
een effectieve diameter. Onze resultaten laten zien dat de diameter van de sphero-
xv
cilinder de beste keuze voor de effectieve diameter is om het warmtetransport te
karakteriseren.
We verwachten dat deze nieuwe resultaten de numerieke modellering van deeltjes-
stromingen m.b.v. multi-scale methoden aanzienlijk zullen verbeteren.
Nomenclature
Variables
A surface area, [m2]
ap specific surface area, [m−1]
Dp particle diameter, [m]
De effective diameter, [m]
Deq equivalent diameter, [m]
Ds Sauter mean diameter, [m]
cp specific heat, [J / (kg · K)]
F force, [N]
h average heat transfer coefficient , [W / (m2 · K)]
Ip moment of inertia, [kg ·m2]
k thermal conductivity , [W / (m2 · K)]
nx, ny, nz number of cells in x, y, z direction, [-]
Np number of particle, [-]
p pressure, [Pa]
Qp particle-fluid heat transfer, [W]
q heat source term, [W/m3]
r,R radius, [m]
t time, [s]
T fluid temperature, [K]
∆T thermal driving force, [K]
u fluid velocity, [m/s]
Us superficial gas velocity, [m/s]
V volume, [m3]
Vp particle volume, [m3]
x position, [m]
xvii
xviii Nomenclature
Greek letters
α thermal diffusivity, [m2/s]
ε voidage, [-]
µ viscosity, [Pa · s]ν kinematic viscosity, [m2/s]
ρ density, [kg/m3]
φ solid volume fraction, [-]
ωp angular velocity, [rad / s ]
Subscripts and superscripts
b bulk
c cross sectional area
g gas phase
p particle
e effective
s solid
k Lagrangian point
ax thermal dispersion
disp dispersion
x, y, z in x, y, z direction
Abbreviations
CFD Computational Fluid Dynamics
DNS Direct Numerical Simulation
ICCG Incomplete Cholesky conjugate gradient
IB Immersed Boundary
IBM Immersed Boundary Method
LBM Lattice-Boltzmann Method
PDF Probability density function
TFM Two-Fluid Model
DPM Discrete particle Model
HTC Heat transfer coefficient
Dimensionless numbers
Pr Prandtl number, ν/α
Nomenclature xix
Re Reynolds number, ρvD/µ
Nu Nusselt number, hD/k
1
CH
AP
TE
R
Introduction
Abstract
In this chapter a brief introduction to gas-solid flows, with an emphasis on modelling
is given. Non-isothermal gas-solid flows are encountered in a variety of industrial
processes utilizing packed and fluidized bed chemical reactors. In order to arrive at
an optimal design and control of such systems, a precise prediction of the tempera-
ture distribution, as well as the flow field inside the equipment is necessary. In this
thesis, we employ the Immersed Boundary Method (IBM) to study non-isothermal
flows through stationary arrays of (non-)spherical particles. The computed mean heat
transfer coefficient of the bed is compared with well-known heat transfer correlations
for packed and fluidized beds.
1
2 Chapter 1. Introduction
1.1 Background and motivation
Hydrodynamic and thermal interactions between fluid and solid phases are frequently
encountered in a wide range of industrial processes like adsorption columns, heat
regenerators, packed and fluidized beds. Understanding the transport phenomena
that prevail in such systems is of utmost importance to improve performance and
facilitate optimal design of these systems.
During the past decades, extensive experimental and theoretical investigations
have been conducted to characterize heat transfer in multiphase systems. It is gen-
erally accepted that the thermal behavior of a bed can be described in terms of an
average heat transfer coefficient (HTC). The HTC is usually obtained from experi-
mental data combined with a simple interpretation model that attempts to represent
the thermal behavior of the system. However, transport phenomena in the void spaces
between particles, due to the existence of a wide range of spatial and temporal scales,
can be very complex. Therefore, selection of a proper thermal model for interpreting
the experimental data is not straightforward.
Extensive studies, as reviewed by Botterill (1975), Gunn (1978), Wakao et al.
(1979) and Kunii and Levenspiel (1991), have been published in this field and many
empirical correlations have been proposed to characterize the HTC in packed and
fluidized beds. These correlations cover a wide range of systems (e.g. involving sta-
tionary or moving particles) and operating conditions (e.g. a wide range of Reynolds
numbers, particle shapes and particle arrangements). The variety of parameters can
affect the heat transfer significantly in these particulate systems. For this reason, a
significant scatter of the experimental HTC is observed in packed and fluidized beds.
This scattering mainly originates from uncertainty and inconsistency among the
experimental data and different assumptions employed in the models used for the
interpretation of the data. All these factors lead to ambiguity in heat transfer cor-
relations in multiphase systems. Therefore it is difficult to suggest one “universal”
correlation with confidence. Therefore, further studies are required to verify the heat
transfer correlations in multiphase systems (at least for specific operating conditions)
and employed assumptions.
1.2 Computational Fluid Dynamics
Recently, with the increase of computational power, numerical simulation is con-
sidered as a promising tool for prediction of the multiphase system behavior. In
Computational Fluid Dynamics (CFD) the actual packing geometry is taken as in-
put whereas the detailed flow and heat transfer patterns are produced as output.
However, simulation of all details of a multiphase system of industrial size is compu-
tationally expensive (due to the existence of a wide range of spatial scales). For this
reason, multi-scale approaches have been adopted (van der Hoef et al. (2008)).
1.3. Multi-scale modelling 3
1.3 Multi-scale modelling
The well-known CFD techniques for simulation of particulate flows proposed in the
literature can be classified in three groups: Two-Fluid model; the interaction between
the phases is approximated by empirical correlations), Discrete Element method (the
fluid motion is described by the Navier-Stokes equations whereas each particle is
described in terms of Lagrangian equations of motion; the interaction between the
phases is represented with empirical correlations) and Direct Numerical Simulations
(the fluid and particulate phases are treated by considering the Navier-Stokes equa-
tion and the Lagrangian equations of motion, respectively; the interaction between
the phases is enforced through the no-slip boundary condition at the surface of the
particle)(Fig 1.1).
At present the Two-Fluid and Discrete Element methods are most widely used
to study the local gas-solid flow structure and thermal behavior of particulate flows.
Although they are computationally less demanding than DNS, they suffer from uncer-
tainties of the boundary conditions for the particulate phase and, as indicated above,
from the limitations that are inherent in using the empirical correlations. With the
rapid increase of computational power, DNS has received considerable attention for
the detailed simulation of particulate flows. DNS methods basically fall into two
classes: boundary-fitted and non-boundary-fitted approaches. In the boundary-fitted
approach, e.g., the arbitrary Lagrangian-Eulerian method Feng et al. (1994), the
flow is solved on a boundary-fitted mesh, and thus re-meshing is required when the
particles move.
Non-boundary-fitted methods, such as the Immersed Boundary method (IB) Pe-
skin (1977), employ a fixed Cartesian mesh for the fluid and moving Lagrangian points
for the particles. In general the non-boundary-fitted methods are more efficient than
the boundary-fitted methods since no remeshing is required. The IB method was first
proposed by Peskin (1977) for simulation of systems with a moving complex bound-
ary. In this method an IB forcing term is introduced into the momentum equation
to describe the mutual interaction between the IB and the fluid. Then this force
is distributed to the Eulerian grid to enforce the no-slip boundary condition on the
IB. Various formulations of the IB forcing have been derived so far. Goldstein et al.
(1993) and Saiki and Biringen (1996) proposed a scheme in which the IB forcing is
a feedback on the difference between the calculated velocity and the desired veloc-
ity. The main drawback of this scheme is that it contains parameters that must be
tuned. Mohd-Yusof (1997) introduced a direct forcing scheme to calculate the in-
teraction force between IB and fluid, which requires no parameters. In this method
the IB forcing term is set by the difference between the interpolated velocity on the
Lagrangian point and the desired boundary velocity. Uhlmann (2005), and later Feng
and Michaelides (2009), combined the advantages of a direct forcing scheme with the
IB method to study the particulate flows in multiphase system.
4 Chapter 1. Introduction
Figure 1.1: Gas-solid multi-scale modelling hierarchy with left: Fully resolved Di-rect Numerical Simulations, center: Discrete Element method and right: Two-FluidModel. The detail of the simulation decreases from left to the right.
1.4 Thesis outline
The objective of this study is to investigate non-isothermal particulate flows using
DNS for a wider range of Reynolds numbers and solids volume fractions.
In Chapter 2 the definitions of HTCs are presented. Then the experimental and
numerical methods for heat transfer characteristics inside packed and fluidized beds
are discussed. A selection of heat transfer correlations are examined and the thermal
models are discussed.
In Chapter 3 the IB method proposed by Uhlmann for DNS of fluid flow through
dense fluid-particle systems is extended to systems with interphase heat transport.
Forced convection heat transfer was simulated for a single sphere and an in-line array
of 3 spheres to assess the accuracy of the present method.
In Chapter 4 non-isothermal flows through stationary arrays of monodisperse
spherical particles are studied. The computed mean HTC of the bed is compared
with well-known heat transfer correlations for packed and fluidized beds. Moreover,
the deviation of the particle HTC from the average value is quantified. In addition the
1.4. Thesis outline 5
effect of heterogeneity on the fluid-particle heat transfer coefficients is investigated as
well.
In Chapter 5 bidisperse random arrays of spheres are investigated to obtain
HTCs for a range of mass fraction and solids volume fractions.
In Chapter 6 Direct numerical simulations are conducted to characterize the
fluid-particle HTC in fixed random arrays of non-spherical particles. In this study
sphero-cylinders were chosen. The simulations are performed for different solids vol-
ume fractions and particle sizes over low to moderate Reynolds numbers. According
to the detailed heat flow pattern, the average HTC is calculated in terms of the
operating conditions.
In Chapter 7 the contributions of the thesis are summarized and some recom-
mendations are provided for future work.
2
CH
AP
TE
R
Heat transfer in fluid-particle systems
2.1 Abstract
This chapter presents firstly the definitions of the variables used in this thesis. Then
an overview of the few well-known heat transfer correlations that are currently used
in engineering applications is provided. The experimental methods of measuring heat
transfer characteristics in packed and fluidized beds are discussed briefly. At the end,
the numerical model used in this work is introduced.
7
8 Chapter 2. Heat transfer in fluid-particle systems
2.2 Introduction
During the past decades, extensive experimental and theoretical investigations have
been conducted to characterize heat transfer in multiphase systems. In these type of
systems heat can be transported by 3 different modes: conduction (e.g. wall-particle,
particle-particle, within fluid etc), convection (e.g. fluid-particle, wall-fluid etc) and
radiation (e.g. wall-particle, particle-particle, particle-fluid etc). It is generally ac-
cepted that each of these modes in a bed can be described in terms of an average
HTC. It is common to represent the overall heat transfer as a sum of all the modes
of heat transfer. In this thesis, we limit our discussion to convective heat transfer
between particle and fluid.
The HTC is usually obtained from experimental data combined with a simple
interpretation model that attempts to represent the thermal behavior of the system.
However, transport phenomena in the void spaces between particles, due to a wide
range of spatial and temporal scales, can be very complex. Therefore, selection of a
proper thermal model for interpreting the experimental data is not straightforward.
In this chapter the HTC is introduced and some convective HTC definitions for
packed and fluidized bed adopted in literature are discussed. In addition some heat
transfer correlations of packed and fluidized beds are presented and the thermal mod-
els are discussed.
2.3 Heat transfer coefficients in a random array of
particles
In this section we will discuss several HTC definitions, exact and approximate, used
in this study. The HTC for creeping flow past an ordered bed of particles can be
obtained from theoretical studies. On the other hand in a heterogeneous system, due
to absence of a uniform solids volume fraction, φ, and flow field as well, there exists
a considerable variation of the HTC in the bed. For heterogeneous systems at finite
Reynolds numbers we need to resort to numerical simulations.
Characterization of heat transfer in a heterogeneous system in terms of the local
micro-structural data is a difficult task. Between these two limiting cases of homo-
geneously structured and heterogeneous variation on larger length scales we find the
so called statistically homogeneous system. In this system the local φ in one part
of the system is the same to the other parts. Although the particles are distributed
randomly in this system, the statistical properties of the system (e.g. the local φ in
a plane) are uniform (Fig 2.1).
To define a HTC we need first to define a suitable temperature difference between
the particles and the fluid. In this study we will assume that all particles will be
kept at a constant fixed temperature Ts. When formulating macro balances for en-
ergy transport in a stationary system the cup-averaged temperature appears in the
2.3. Heat transfer coefficients in a random array of particles 9
In this study, we employ the Immersed Boundary Method (IBM) to study non-isothermal flows throughstationary arrays of spherical particles. The computed mean HTC of the bed will be compared with well-knownheat transfer correlations for packed and fluidized beds. Next, we will characterize the deviation of particleHTC’s from the average value in terms of operation conditions. The paper is organized as follows. First, thegoverning equations and numerical solution method are discussed. Then the HTCs of a fixed bed of particlesare compared with well-known correlations. The effect of heterogeneity on the particle heat flux is investigatedas well. Last, the summary and conclusion are given in the final section.
3 Heat transfer coefficients in a random array of particles
In this section we will discuss several HTC definitions, exact and approximate, used in this study. The HTC forcreeping flow past an ordered bed of particles can be obtained from theoretical studies. On the other hand ina heterogeneous system, due to absence of a uniform solids volume fraction, φ, and fluid velocity field as well,there is a considerable variation in the bed. For heterogeneous systems at finite Reynolds numbers we need toresort to numerical simulations.
Characterization of heat transfer in a heterogeneous system in terms of local microstructural detail is adifficult task. Between these two limiting cases of statistically homogeneously structured and heterogeneousvariation on larger length scales we find the so called statistically homogeneous system. In this system the localφ in one part of the system is the same to the other parts. Although the particles are distributed randomly inthis system, the statistical properties of the system (e.g. the local φ in a plane) are uniform (Fig 1).
(a) (b)
Figure 1: A schematic of a) ordered b) statistically homogeneous (squares represent two subregions with thesame solid volume fraction).
To define a HTC we need first to define a suitable temperature difference between the particles and the fluid.In this study we will assume that all solid particles will be kept at a constant fixed temperature Ts. Whenformulating macro balances for energy transport in a stationary system the cup-averaged temperature appearsin the equations. Because we are also considering stationary systems it is most convenient to use a cup-averagedtemperature to characterize the gas phase bulk temperature,
Tb(x) =
∫Acux(x, y, z)T (x, y, z) dydz∫
Acux(x, y, z) dydz
(1)
where Ac is the cross sectional area of the domain. Here the flow direction is taken to be x. It is reasonableto expect that, for flow in the x direction, the temperature is statistically homogeneous in the perpendiculardirections and increases toward Ts along the x-axis. We will consider the heating of cold gas by hot particlesso the convenient driving force at position x is Ts − Tb(x). In section 6.1 we will also consider a more locallydefined temperature difference as thermal driving force.
3
(a)
In this study, we employ the Immersed Boundary Method (IBM) to study non-isothermal flows throughstationary arrays of spherical particles. The computed mean HTC of the bed will be compared with well-knownheat transfer correlations for packed and fluidized beds. Next, we will characterize the deviation of particleHTC’s from the average value in terms of operation conditions. The paper is organized as follows. First, thegoverning equations and numerical solution method are discussed. Then the HTCs of a fixed bed of particlesare compared with well-known correlations. The effect of heterogeneity on the particle heat flux is investigatedas well. Last, the summary and conclusion are given in the final section.
3 Heat transfer coefficients in a random array of particles
In this section we will discuss several HTC definitions, exact and approximate, used in this study. The HTC forcreeping flow past an ordered bed of particles can be obtained from theoretical studies. On the other hand ina heterogeneous system, due to absence of a uniform solids volume fraction, φ, and fluid velocity field as well,there is a considerable variation in the bed. For heterogeneous systems at finite Reynolds numbers we need toresort to numerical simulations.
Characterization of heat transfer in a heterogeneous system in terms of local microstructural detail is adifficult task. Between these two limiting cases of statistically homogeneously structured and heterogeneousvariation on larger length scales we find the so called statistically homogeneous system. In this system the localφ in one part of the system is the same to the other parts. Although the particles are distributed randomly inthis system, the statistical properties of the system (e.g. the local φ in a plane) are uniform (Fig 1).
(a) (b)
Figure 1: A schematic of a) ordered b) statistically homogeneous (squares represent two subregions with thesame solid volume fraction).
To define a HTC we need first to define a suitable temperature difference between the particles and the fluid.In this study we will assume that all solid particles will be kept at a constant fixed temperature Ts. Whenformulating macro balances for energy transport in a stationary system the cup-averaged temperature appearsin the equations. Because we are also considering stationary systems it is most convenient to use a cup-averagedtemperature to characterize the gas phase bulk temperature,
Tb(x) =
∫Acux(x, y, z)T (x, y, z) dydz∫
Acux(x, y, z) dydz
(1)
where Ac is the cross sectional area of the domain. Here the flow direction is taken to be x. It is reasonableto expect that, for flow in the x direction, the temperature is statistically homogeneous in the perpendiculardirections and increases toward Ts along the x-axis. We will consider the heating of cold gas by hot particlesso the convenient driving force at position x is Ts − Tb(x). In section 6.1 we will also consider a more locallydefined temperature difference as thermal driving force.
3
(b)
Figure 2.1: A schematic of a) ordered b) statistically homogeneous (squares representtwo subregions with the same solid volume fraction).
equations. Because we are also considering stationary systems it is most convenient
to use a cup-averaged temperature to characterize the gas phase bulk temperature,
Tb(x) =
∫Acux(x, y, z)T (x, y, z) dydz∫Acux(x, y, z) dydz
(2.1)
where Ac is the cross sectional area of the domain. Here the flow direction is taken
to be x-direction. It is reasonable to expect that, for flow in the x-direction, the
temperature is statistically homogeneous in the perpendicular directions and increases
toward Ts along the x-axis. We will consider the heating of cold gas by hot particles
so a convenient driving force at position x is Ts − Tb(x).
10 Chapter 2. Heat transfer in fluid-particle systems
For comparison with simplified models we define a rigorous HTC h(x) as follows,
h(x) =Q(x)
Ts − Tb(x)(2.2)
The units of Q(x) is W/m2 and it can be computed as follows: Consider an interval
along the x-axis (which is the flow direction) of width ∆x. This defines a slab of
width ∆x and cross-sectional area Ac. Now consider all parts of the sphere’s surfaces
within this slab. In this case Q(x) is the heat flow rate from the particles to the
gas through all these surface parts divided by the total area of these parts. In the
continuum limit one would take ∆x→ 0, in a computation one uses a finite ∆x. For
a statistically homogeneous system and fully developed flow it is expected that h(x)
is in fact independent of x.
The classical one-dimensional model for thermal behavior of packed and fluidized
beds has been proposed by Schumann (1929). In this model the convective heat
exchange between solid and fluid is assumed to be equal to the change in internal
energy of the fluid. It can be obtained by a macro balance and neglecting dispersion
effects
hno−disp ap (Ts − Tb(x)) = U ρcpdTb(x)
dx, (2.3)
here ap is the specific surface area of the particles, i.e., ap = (6φ)/Dp, ρ is the fluid
density, cp is the heat capacity and U is the superficial gas velocity. The solution of
this equation gives,
− 1
xln
[Ts − Tb(x)
Ts − Tb(0)
]=hno−disp apU ρcp
, (2.4)
Experimentally this is a very convenient result because cup-mixing inlet and outlet
temperature are readily available. This information can be used to estimate hno−disp.
If dispersion effects are negligible then the ‘real’ heat transfer coefficient, h(x), is
expected to be close to hno−disp.
However, according to Gunn (1978) and Wakao et al. (1979), thermal dispersion
must be considered in the thermal model for accurate predictions at Re < 100. The
neglect of thermal dispersion effect and the accuracy of αax can significantly affect the
interpretation of experimental results. It is observed that the reported HTCs by some
researches differ from others by several orders of magnitude at the same operating
conditions when the Reynolds number is lower than 100. The detailed explanation of
this issue was provided in Gunn (1978).
If the thermal diffusion term is included in the Schumann model, the continuous
solid phase model (Wakao et al. (1979)) is found:
hdisp ap (Ts − Tb(x)) = ρcp
(UdTb(x)
dx− αax
d2 Tb(x)
dx2
)(2.5)
2.4. Experimental methods 11
where αax is the axial fluid thermal dispersion coefficient and must be estimated
experimentally or theoretically. The solution of this equation is,
hdisp apρcp
= U λ− αaxλ2, with λ = − 1
xln
[Ts − Tb(x)
Ts − Tb(0)
]. (2.6)
Because λ > 0 this shows that, when inferring heat transfer coefficients from inlet and
outlet temperatures one finds that, hno−disp > hdisp. The exact numerical estimation
of the αax is beyond the scope of this thesis but extensive studies can be found in
literature that investigate αax in packed and fluidized beds (e.g. Gunn and Souza
(1974) and Kuwahara et al. (1996)).
In DNS Tb can be readily obtained along the bed, so hno−disp can be computed
directly form the simulation. When dispersion effects would be correctly modelled
hdisp is expected to be close to the ‘real’ HTC, h(x). Instead of modeling the disper-
sion effects we directly compare h(x) and hno−disp and assume that the difference is
mainly caused by the neglect of dispersion effects.
The third type of HTC we will consider is the HTC per particle,
hp =1
Ap
QpTs − Tb(xp)
. (2.7)
In this definition the total heat flow from a particle to the gas is divided by the
surface area of the particle. The gas temperature is the cup-averaged temperature
evaluated at the x position of the center of the particle. This heat transfer coefficient
can be computed for each particle individually. By definition, when averaging over
all particles in a slab around a position x, the average hp should be close to h(x). By
monitoring the variation of hp from particle to particle we are able to quantify the
influence of heterogeneity.
We report our results for the HTCs in a dimensionless way by means of Nusselt
numbers, defined as Nu = hDp/k, where k is the thermal conductivity of the gas.
2.4 Experimental methods
Experimental characterizations of HTCs for a wide variety of fluid-solid systems have
been conducted using various experimental techniques, under either steady-state or
unsteady-state conditions. An extensive review on experimental works on particle-
fluid heat transfer can be found in Wakao et al. (1979).
The typical experimental setup used to characterize the particle-fluid HTC in a
packed bed is shown schematically in Fig. 2.2. It is composed of an inlet section, a
packed section, an outlet section and an assembly of thermocouples to measure the
fluid temperature at the inlet of the bed and the bed exit. According to the measured
temperatures, the HTC is obtained by solving the inverse problems using one of the
various thermal models. The simplest thermal model is one-dimensional and contains
12 Chapter 2. Heat transfer in fluid-particle systems
a convective HTC based on the difference between the particle temperature and the
local cup-mixing temperature of fluid (Eq. 2.4). The most common type of a packed
bed consist of a random distribution of particles in a cylindrical tube. In many cases,
the particles are either spherical or cylindrical. Recently, structured packed beds have
also been studied (Yang et al. (2010)). It was shown that the overall heat transfer
performance will be significantly improved in a structured packed bed. In this study,
only random fixed arrays of particles are investigated.
The HTC in a fluidized bed can also be measured by injection of a hot (or cold)
freely moving particle in a cold (or hot) fluidized bed (Fig. 2.3) (Parmar and Hayhurst
(2002),Collier et al. (2004),Scott et al. (2004) etc). The temperature of the active
particle is measured by a embedded thermocouple wire at the center of particle.
Subsequently, the temperature of the particle against time can be measured. From
this dynamic temperature response of the mobile particle, the particle-fluid HTC in
the fluidized bed can be obtained. This technique given results for a packed bed, when
the superficial velocity of the fluid is reduced below that for incipient fluidization.
Wu and Hwang (1998) constructed a fixed bed with by stringing thread trough the
spherical particles in the bed. The solids volume fraction of the bed can be adjusted
by changing the particle size and the space between the particles (Fig. 2.4). From
these experiments the convective HTC can be determined as a function of Reynolds
number and solids volume fraction.
0 10 20 30 40 50 60 70 80 90 1000
5
10
15
20
25
Reynolds number
Nuss
elt
No.
Gunn (φ = 0.6)
Gunn (φ = 0.5)
Gunn (φ = 0.4)
Gunn (φ = 0.3)
Gunn (φ = 0.2)Wakao
Figure 1: The mean Nusselt number in a packed bed obtained from Gunn and Wakao correlation and numericalsimulations. (φ = 0.6)
Thermocouple
Outlet section
Thermocouple
Inlet section
Packed section
Flow of the fluid
2Figure 2.2: Schematic representation of packed bed heat transfer measurements.
2.4. Experimental methods 13
Figure 2.3: Schematic representation of fluidized bed heat transfer measurements.
Thermocouple
Outlet section
Thermocouple
Inlet section
Packed section
Flow of the fluid
4
Figure 2.4: Schematic representation of the setup of heat transfer experiment usedby Wu and Hwang (1998).
14 Chapter 2. Heat transfer in fluid-particle systems
2.5 Heat transfer correlations
A large number of empirical or semi-empirical correlations have been developed to
characterize the fluid-particle heat transfer in packed and fluidized beds. Some studies
on the heat transfer in packed beds reveal that the fluid-particle HTC for spherical
particles can be predicted reasonably well from the correlations of Wakao et al. (1979):
NuWakao = 2 + 1.1Re0.6Pr1/3 (2.8)
and Gunn (1978):
NuGunn = (7− 10ε+ 5ε2)(1 + 0.7Re0.2Pr1/3) + (1.33− 2.4ε+ 1.2ε2)Re0.7Pr1/3
for 0 < Re < 105, and 0.35 < ε < 1 (2.9)
where ε is voidage of the bed (ε = 1 − φ). Wakao did not explicitly state whether
Eq. 2.8 is valid only for a specific solids volume fraction or for any value of this
parameters. However, it is common to employ Eq. 2.8 in packed beds (φ ≈ 0.6).
Fig. 2.5 shows the Nusselt number plotted against the Reynolds number at differ-
ent solids volume fractions when the Prandtl number is 1. The correlation by Gunn
produces higher Nusselt numbers (φ = 0.6) than that by Wakao. The deviations
between these correlations are 80% and 25% when Reynolds numbers are 10 and 100,
respectively.
Although heat correlations have been used successfully for design purposes, these
are not generally applicable for different systems and a wider range of operating
conditions. Beside these issues, the correlations have limitations in providing insight
into the complex thermal dynamics. For example, the contribution of each of the three
basic heat transfer mechanisms (convection from fluid, conduction from particles and
radiation) to the total HTC is difficult to determine. Moreover, the HTC depends
strongly on the gas-solid flow pattern. Also the effect of micro-structural information
on the HTC is not straightforward and can not be quantified easily with an empirical
heat transfer correlation. In other words, the empirical correlations only provide a
description of the average thermal dynamic behavior of a system.
2.6 DNS of fluid-particle heat transfer
With the rapid increase of computational power, DNS has attracted considerable in-
terest for the simulation of particulate flows. Although fully resolved simulation of
a large multiphase system is not feasible, the DNS of a small system can increase
our insight into transport phenomena in complex multiphase systems. In the DNS
approach, the fluid and solid phases are treated by considering the Navier-Stokes
equations and the Newtonian equations of motion, respectively. The mutual interac-
tions between the phases are obtained by enforcing the appropriate boundary condi-
tions at the surface of the particle (e.g. no-slip and Dirichlet boundary condition for
2.6. DNS of fluid-particle heat transfer 15
Figure 2.5: Comparisons of Nusselt number correlations of Gunn and Wakao.
16 Chapter 2. Heat transfer in fluid-particle systems
momentum and heat transfer, respectively) and therefore no empirical correlations
are required. Consequently, the drag and heat transfer coefficients can be deter-
mined from DNS results. With this outstanding advantage of DNS the accuracy of
well-known correlations can be assessed with DNS results obtained from well-defined
systems.
All of these facts motivate us to investigate and compare experimental and nu-
merical thermal models for packed and fluidized beds in more detail with the help of
DNS. The results of this study enable us to improve the prediction of the numerical
simulation of heat transfer in particulate systems.
2.6.1 Physical model
Fig. 2.6 shows a schematic of the computational domain used in a DNS. The par-
ticle configuration was obtained by randomly distributing N = 54 non-overlapping
spherical particles in a 3-dimensional duct by a standard Monte Carlo method. The
volume of the packed section is obtained as the ratio of total volume of particles to
the desired solids volume fraction (V = L3 = NπD3p/(6φ) ). The sizes of inlet and
outlet sections are equal and depend on the particle size.
The spheres are maintained at a constant temperature Ts and exchange heat with
the cold flowing fluid with a constant inflow temperature T0. The fluid flows through
the duct in streamwise direction (x direction). Periodic boundary conditions are im-
posed in spanwise directions (y and z directions) in order to remove wall effects. Two
different types of boundary conditions can be applied in streamwise direction. In
the first approach, the periodic and similarity boundary conditions are applied for
velocity and temperature, respectively (e.g. Kuwahara et al. (1996) and Tenneti et al.
(2013)). The second approach consists of imposing inlet and outlet boundary condi-
tion. Since the similarity conditions can not be imposed in reality, such a physical
model does not correspond to a real experimental system. In experiments, the fluid
enters the bed with uniform velocity and temperature. Due to the sudden change
in the solids volume fraction at the inlet and outlet of a bed, the local crosssection-
averaged HTC is not constant along the bed and is affected by the entrance effect in
experiments. In order to analyze the entrance effect in a simulation, a second type
of boundary condition is introduced. In this approach, the computational domain is
a 3-dimensional duct that is divided into 3 sections: inlet, packed and outlet. The
non-overlapping spherical particles are distributed in the packed section. Therefore,
only the packed section is active in heat transfer. In this study, we employ the second
approach and investigate the effect of the entrance region on the HTC of the bed.
However, we remove the effect of the entrance region when the average bulk HTC is
reported.
The boundary conditions for this simulation were typically set as follows:
• At the inlet (x = 0), uniform axial velocity U and temperature T0 of the fluid
2.6. DNS of fluid-particle heat transfer 17
Packed section Outlet sectionInlet section
Figure 2: A typical particle configuration used in the simulations for φ=0.1.
• At the outlet (x = X), the boundary conditions are:
∂~u
∂x= 0,
∂T
∂x= 0. (12)
• On the periodic boundaries:
~u(x, 0, z) = ~u(x, Y, z), T (x, 0, z) = T (x, Y, z)
~u(x, y, 0) = ~u(x, y, Z), T (x, y, 0) = T (x, y, Z)(13)
where Y and Z are the size of domain in y and z directions, respectively.
• At the particle surface, Dirichlet conditions were used for the velocity (no-slip) and temperature,
~u = 0, T = Ts. (14)
4.2 Numerical method
The DNS of this system had been carried out by using the Immersed Boundary (IB) method. This method hasbeen discussed in detail in Tavassoli et. al. (2013). Here we provide a brief outline of the method.
In IB methods, a fixed Eulerian grid and moving Lagrangian boundary points are employed for modelingthe fluid and particle phases. The fluid occupies the full domain, even inside the particles, and is modeled onthe fixed Eulerian grid. The location of the particle is specified by Lagrangian points that are distributed overthe outer boundary of the particle. The mutual hydrodynamic and thermal interactions between the fluid andthe particle ( ~fIB and QIB) are calculated iteratively such that the desired hydrodynamic and thermal boundaryconditions are satisfied on the immersed boundary (i.e no-slip velocity and Dirichlet boundary conditions). Thefinite difference method was used to discretize the momentum and energy equation on the staggered Euleriangrid. The IB source terms are first calculated at the Lagrangian points, then these source terms are distributedto the neighbor Eulerian grid nodes by means of regularized delta-functions. ~fIB and QIB are non-zero only overthe region near to IB interface. The same delta functions is used to estimate interpolated values of the velocitiesand temperatures at the Lagrangian points form the Eulerian grids. If the these velocity and temperature donot fulfill the desire boundary condition, the new IB source terms will be recalculated. In other words, the IBsource terms are calculated through an iterative procedure.
The grid independency of solutions has been investigated by performing simulations using maximum 4different grid sizes equal to dp/20, dp/30, dp/40 and dp/50 for each case. The total heat fluxes through of abed obtained by using different mesh sizes were compared with each other. When the results show the desiredconvergence (i.e. when the deviation is lower than 2%), the proper mesh size is selected. In general, the propermesh size is function of Reynolds number and φ. Table 1 reports the employed mesh size for each case in oursimulations.
6
Figure 2.6: A typical particle configuration used in the simulations for φ=0.1.
are imposed:
uy = uz = 0, ux = U, and T = T0. (2.10)
• At the outlet (x = X), the boundary conditions are:
∂u
∂x= 0,
∂T
∂x= 0. (2.11)
• On the periodic boundaries:
u(x, 0, z) = u(x, Y, z), T (x, 0, z) = T (x, Y, z)
u(x, y, 0) = u(x, y, Z), T (x, y, 0) = T (x, y, Z)(2.12)
where Y and Z are the size of domain in y and z directions, respectively.
• At the particle surface, Dirichlet conditions were used for the velocity (no-slip)
and temperature,
u = 0, T = Ts. (2.13)
18 Chapter 2. Heat transfer in fluid-particle systems
2.6.2 Numerical method
All DNS in this thesis had been carried out by using the Immersed Boundary Method.
This method can be used efficiently in non-isothermal complex geometry problems
(such as a random array of particles). The Immersed Boundary Method will be
discussed in detail in the next chapter.
3
CH
AP
TE
R
The Immersed Boundary Method
3.1 Abstract
The IB method proposed by Uhlmann for DNS of fluid flow through dense fluid-particle
systems is extended to systems with interphase heat transport. A fixed Eulerian grid is
employed to solve the momentum and energy equations by traditional computational
fluid dynamics methods. Our numerical method treats the particulate phase by intro-
ducing momentum and heat source terms at the boundary of the solid particle, which
represent the momentum and thermal interactions between fluid and particle. Forced
convection heat transfer was simulated for a single sphere and an in-line array of 3
spheres to assess the accuracy of the present method. All results are in satisfactory
agreement with experimental and numerical results reported in literature.
19
20 Chapter 3. The Immersed Boundary Method
3.2 Introduction
In IB methods, a fixed Eulerian grid and moving Lagrangian boundary points are
employed (Fig. 3.1a). The fluid is assumed to occupy the full domain, even inside the
particles. The particles are represented by Lagrangian points which are distributed
over the outer boundary of the particles (Fig. 3.1b). In this method, IB forces and
energy source terms are introduced into the momentum and thermal energy equations,
respectively, to describe the mutual hydrodynamic and thermal interactions between
the particle and the fluid. The source terms are evaluated iteratively such that the
desired boundary conditions are fulfilled on the IB.
Although the extension of IB method to heat transfer problem is relatively straight-
forward, only few computational results have been published yet. Kim and Choi
(2000) proposed an IB finite-volume method for heat transfer in complex geometries
for fixed particles. Feng and Michaelides (2009, 2008) developed an IB fully explicit
finite-difference method for heat transfer in particle laden flows. Wang et al. (2009)
developed a direct forcing IB procedure called the “multi-direct heat source scheme”.
In this method the IB forcing terms are calculated with the help of an iterative
procedure to enforce the Dirichlet boundary condition at the immersed boundary.
Recently Deen et al. (2012) applied the IB method to study the HTC of dense non-
isothermal fluid-particle systems and compared these with experimental correlations
for a random array of particles with porosity=0.7 and Reynolds number = 36, 72, 108
and 144. That study is the first numerical validation of one of the most well known
heat correlations namely the Gunn correlation. However, the numerical results are
obtained for only one porosity = 0.7 and one Prandtl number = 0.8.
The objective of this chapter is to investigate non-isothermal particulate flow using
a direct forcing IB method. In this method the momentum and energy equations are
solved in the whole domain, including the regions that are occupied by the particles.
The momentum and heat exchange between the phases is accounted for by introducing
momentum and energy source terms in the governing equations. The source terms
are evaluated iteratively such that the velocity and temperature boundary conditions
on the IB are satisfied.
The method is validated by i) comparing the temperature profiles for well-defined
heat conduction problems with the available analytical solutions, and ii) comparing
the results for the convective heat transfer coefficient for flows past a single sphere
and an in-line array of three spheres with data published in literature.
The chapter is organized as follows. First, the governing equations of momentum
and heat transfers in the particulate flow are introduced. Then the numerical solution
method is discussed. Some numerical experiments are conducted to validate the
accuracy of the present method. The summary and conclusion are given in the final
section.
3.2. Introduction 21
4Vk
(a) (b)
Figure 1: Illustration of IB (a) A two-dimensional staggered Cartesian gridwith IB. Locations of ~ux and ~uy are represented by horizontal and verticalarrows ( ,), respectively. Pressure and temperature are positioned at thecentre of each cell ( ). Lagrangian points on IB are shown with filled cir-cles ( ). 4Vk is a volume that is assigned to each Lagrangian point. (b)Representation of a sphere by Lagrangian points.
1
(a)
4Vk
(a) (b)
Figure 1: Illustration of IB (a) A two-dimensional staggered Cartesian gridwith IB. Locations of ~ux and ~uy are represented by horizontal and verticalarrows ( ,), respectively. Pressure and temperature are positioned at thecentre of each cell ( ). Lagrangian points on IB are shown with filled cir-cles ( ). 4Vk is a volume that is assigned to each Lagrangian point. (b)Representation of a sphere by Lagrangian points.
1
(b)
Figure 3.1: Illustration of IB (a) A two-dimensional staggered Cartesian grid withan IB. Locations of ux and uy are represented by horizontal and vertical arrows,respectively. Pressure and temperature are positioned at the center of each cell (filledsquares). Lagrangian points on IB are shown with filled circles. (b) Representationof a sphere by Lagrangian points.
22 Chapter 3. The Immersed Boundary Method
3.3 Mathematical formulation
3.3.1 Governing equations for fluid flow
In the IB method, the governing equations for unsteady incompressible fluid flow with
constant properties and negligible viscous heating effects are:
ρf∂u
∂t+ ρfu · ∇u = −∇p+ µf∇2u + f (3.1)
∇ ·u = 0 (3.2)
∂T
∂t+ u · ∇T = αf∇2T + q (3.3)
In the above equations, ρf , µf and αf are the density, viscosity and thermal diffusivity
of the fluid. u, p and T are the velocity vector field, pressure and temperature of the
fluid, respectively.
In the momentum equation (Eq.3.1), the additional volume forcing term f , com-
pared to the Navier-Stokes equation, is determined such that the velocity boundary
condition is enforced at the fluid-IB interface. Similarly a heat source term q is added
to energy equation (Eq.3.3) to satisfy the temperature boundary condition at the
fluid-IB interface. f and q are non zero only at the IB interface. In fact, f and q are
the mutual momentum and heat exchange, respectively, between the fluid and the
IB.
3.3.2 Determination of IB force term
In the present work, we follow the direct forcing scheme proposed by Uhlmann (2005)
to determine the forcing term that is required to impose a desired velocity ud at the
boundary. In the Uhlmann approach, the forcing term is calculated on a Lagrangian
point instead of the Eulerian grid. The time-discretized form of the momentum
equation Eq. 3.1 at time level n + 1, can be written at the Lagrangian point xk on
the IB as:
ρfun+1k − unk
∆t= RHSuk + Fk (3.4)
where RHSuk contains the discrete representation of convective, viscous and the
pressure gradient terms. Eq.(3.4) can be written as:
Fk = ρfun+1k − unk
∆t− RHSuk = ρf
un+1k − u
(0)k
∆t+ ρf
u(0)k − unk
∆t− RHSuk (3.5)
3.3. Mathematical formulation 23
where u(0)k is a temporary velocity which corresponds to the flow field without the
forcing term:
ρfu
(0)k − unk
∆t− RHSuk = 0 (3.6)
The no-slip boundary condition dictates that un+1k = ud. Therefore, the forcing term
on Lagrangian point xk is evaluated as:
Fk =ρf∆t
(ud − u(0)k ) (3.7)
3.3.3 Determination of IB Heat source term
The desired temperature Td on IB is satisfied by imposing a heat source term Qk,
which is introduced into the time-discretized form of the energy equation Eq.(3.3) at
the Lagrangian point xk:
Tn+1k − Tnk
∆t= RHSTk
+Qk (3.8)
where RHSTkcontains the conduction and convective terms in their discrete form.
Eq.(3.8) can be written as:
Qk =Tn+1k − Tnk
∆t− RHSTk
=Tn+1k − T (0)
k
∆t+T
(0)k − Tnk
∆t− RHSTk
(3.9)
where T 0k is a temporary temperature which satisfies the energy equation without IB
heat source term:
T(0)k − Tnk
∆t− RHSTk
= 0 (3.10)
At the IB interface, the fluid and particle temperatures are equal: Tf= Tn+1k =Td.
Therefore, the IB heat source term is computed using the following equation:
Qk =Td − T (0)
k
∆t(3.11)
3.3.4 Interpolation of quantities between Eulerian and
Lagrangian coordinates
Eqs.(3.7) and (3.11) show that the IB force and heat source terms are calculated
using the velocity uk and temperature Tk at the Langrangian point xk. A so called
regularized delta function D(x,xk) is required to obtain the Lagrangian properties ukand Tk by interpolation of values at the appropriate Eulerian grid points near to the
Lagrangian point xk. The velocity and temperature on the Lagrangian point xk at
the immersed boundary are obtained from:
uk(xk) =∑
xεΩ
u(x)D(x,xk) (3.12)
24 Chapter 3. The Immersed Boundary Method
Tk(xk) =∑
xεΩ
T (x)D(x,xk) (3.13)
where Ω is the supporting domain of D(x,xk).
Furthermore, the effects of IB force Fk(xk) and heat source Qk(xk) that are
located at the Lagrangian point xk are distributed to the nearest Eulerian grid points
by applying the same regularized delta function. This leads to an estimation of the
Eulerian forcing f(x) and heat source q(x) terms at Eulerian grid x and expressed
as:
f(x) =
N∑
k=1
Fk(xk)D(x,xk)∆Vkh3
(3.14)
q(x) =
N∑
k=1
Qk(xk)D(x,xk)∆Vkh3
(3.15)
where N is the number of Lagrangian points, and ∆Vk is the volume that belongs
to Lagrangian point k. Since D(x,xk) is nonzero only in the supporting domain,
f(x) and heat q(x) are obtained from Lagrangian points that are located inside the
supporting domain of D(x,xk).
The choice of D(x,xk) must fulfill certain criteria, as discussed by Peskin (1977).
A variety of functions D(x,xk) has been proposed in the literature. We have analyzed
the accuracy of regularized delta functions by selecting three types of them (Peskin
(1977), Darmana et al. (2007) and Tornberg and Engquist (2004)). No significant
differences were found in the results using these regularized delta functions. Therefore,
we use the regularized delta function introduced by Darmana et al. (2007), since it is
computationally cheaper than the others.
3.3.5 Motion and Energy equations of the particles
Although we consider systems with stationary particles in this work, for completeness
we also describe the equations of motion of the particles in case they are free to move.
The dynamics of a solid particle is governed by the Newtonian equations of motion,
which are given by:
ρpVpdUp
dt= ρpVpg −
∮
∂S
(τ f ·n) ds+
∮
∂S
pn ds (3.16)
Ipdωpdt
= −∮
∂S
((x− xp)× (τ f ·n)) ds (3.17)
3.3. Mathematical formulation 25
τ f = −µf [∇uf + (∇uf )T ].
ρp, Vp, Up, Ip, ωp and xp are the density, volume, translational velocity vector,
moment of inertia, angular velocity vector and the position vector of the center of
mass of the particle; respectively. ∂S represents the surface of the particle. The first
and second terms on the right hand side of Eq.(3.16) are buoyancy force and the
hydrodynamic interaction force between the particle and the fluid, respectively. The
right hand side of Eq.(3.17) is the torque that the fluid exerts on a particle.
According to previous discussion, the hydrodynamic interaction force between a
particle and the fluid can be obtained by summation of IB force terms at the surface of
the particle. The IB force changes both the interior and exterior and as a consequence
the momentum of unphysical internal fluid must be subtracted to obtain the correct
hydrodynamic force between the particle and the fluid. It can be shown (Uhlmann
(2005)) that the inertia of internal fluid is equal to the change of linear momentum
of the center of mass of the fluid inside the particle, hence:
ρfVpdUp
dt= ρfVpg −
∮
∂S
(τ f ·n) ds+
∮
∂S
pn ds+
N∑
k=1
Fk ∆Vk (3.18)
ρfρpIpdωpdt
= −∮
∂S
((x− xp)× (τ f ·n)) ds+
N∑
k=1
(xk − xp)× Fk ∆Vk. (3.19)
By subtracting Eqs.(3.18) and (3.19) from Eqs.(3.16) and (3.17) the following expres-
sions result:
(ρp − ρf )VpdUp
dt= (ρp − ρf )Vpg −
N∑
k=1
Fk ∆Vk (3.20)
Ip(1−ρfρp
)dωpdt
= −N∑
k=1
(xk − xp)× Fk ∆Vk. (3.21)
The transient temperature of the particle can be determined by considering the energy
balance around the particle:
ρpVpcpdTpdt
= kf
∮
∂S
(∇T ·n) ds, (3.22)
where cp and kf are the specific heat of the particle and thermal conductivity of the
fluid. By the same argument proposed for particle motion, the following relationship
can be derived for the energy equation of the particle (at small Biot number):
(ρpcp − ρfcf )VpdTpdt
= −ρfcfN∑
k=1
Qk ∆Vk. (3.23)
26 Chapter 3. The Immersed Boundary Method
3.4 Numerical solution method
The governing equations were solved using a finite difference scheme based on a
staggered Eulerian grid. The above set of equations for the momentum and heat
advection is integrated in time using the fractional-step method. The nonlinear con-
vection terms Cu = u · ∇u and CT = u · ∇T in the momentum and energy equations,
respectively, are treated by the explicit second-order Adams-Bashforth method:
Cn+ 1
2u ≈ 3
2Cnu −
1
2Cn−1u (3.24)
Cn+ 1
2
T ≈ 3
2CnT −
1
2Cn−1T (3.25)
The viscous Su = −µf∇2u and conduction ST = −αf∇2T terms in the mo-
mentum and energy equations, respectively, are discretised in time using the Crank-
Nicolson scheme.
Sn+ 1
2u ≈ 1
2(Sn+1u + Snu) = −1
2
µfh2
L(un+1 + un) (3.26)
Sn+ 1
2
T ≈ 1
2(Sn+1T + SnT ) = −1
2
αfh2
L(Tn+1 + Tn) (3.27)
where L represents the discretised Laplace operator and h the grid size. The proce-
dure for the implementation of this scheme is briefly explained below. The diagrams
detailing the computational sequences are given in Figs. 3.2 and 3.3.
For every time step, first a temporary velocity field is calculated by solving the
momentum equations Eq. (3.1) without the IB force terms. Subsequently, the interpo-
lated velocity at each Lagrangian point is calculated using Eq. (3.12). Subsequently,
the IB force term is obtained via Eq. (3.7) whereafter the Lagrangian IB force term
is distributed to the neighboring Eulerian grid points according to Eq. (3.14). Next,
the new flow field is calculated from the momentum equations including the IB force
term. If the error between the calculated velocity at the IB and the desired veloc-
ity exceeds a pre-defined threshold an iterative procedure is employed, similar as in
Wang et al. (2009). The iterative procedure is performed until the no-slip boundary
condition is obeyed within a pre-defined error limit or after a maximum number of
iterations. At the end of the iterative processing, the new velocity field un+1 and
pressure field pn+1 are obtained according to Eqs. (3.1) and (3.2).
The same algorithm is applied for the calculation of the temperature field. By
solving the energy equation Eq. (3.3) without IB heat source term, the temporary
temperature field of the whole domain is obtained. Eq. (3.13) is employed to obtain
an interpolated temperature at the IB, which is used to calculate the heat source term
3.5. Verification 27
according to Eq. (3.11). The IB heat source term is distributed (see Eq. (3.15)) to the
Eulerian grid to obtain the Eulerian heat source term from which the new intermediate
temperature field is calculated implicitly. If the error between the temperature on
the IB and the desired temperature is larger than a pre-defined threshold, the heat
source term is estimated again. The iterative procedure is continued until a specified
convergence criterion is satisfied.
It must be noted that since the physical properties of the fluid are considered to be
constant in this work, the momentum and energy equations are decoupled. Therefore,
first the momentum equations are solved and subsequently the temperature field is
obtained. However, extension of this algorithm to coupled momentum and energy
equations can be readily achieved by solving the momentum and energy equations
simultaneously.
3.5 Verification
The proposed numerical method is validated first by comparing the numerical results
for the heat conduction around a stationary spherical particle immersed in an infi-
nite stationary fluid with the analytical solution. In addition the computed Nusselt
number for forced convection around a hot spherical particle is compared to well-
known empirical correlations. The effect of neighbouring particles is examined for a
linear array of three spherical particles and computed heat transfer coefficients are
compared with the results reported by Maheshwari et al. (2006).
3.5.1 Cooling of a sphere in contact with an unbounded fluid
A solid sphere of radius R at a constant temperature Ts is suddenly immersed at
time t = 0 in unbounded fluid of temperature T∞. The radial distribution of the
fluid temperature T (r, t), r > R, follows from the heat diffusion equation in spherical
coordinates:
∂T (r, t)
∂t=
α
r2
∂
∂r
(r2 ∂T (r, t)
∂r
)(3.28)
where α is thermal diffusivity. The analytical solution of Eq. 3.28 is:
T (r, t)− T∞Ts − T∞
=R
r
(1− erf
(r −R√
4αt
)). (3.29)
The Nusselt number, defined as Nu = hfDp/kf ,where hf , Dp and kf represent
the heat transfer coefficient, particle diameter and thermal conductivity of the fluid,
respectively, for the cooling of a sphere is then:
NuAnalytical(t) = 2 +2√π
R√αt. (3.30)
We consider a geometry in which the sphere is positioned at the center of a cubic
computational domain with edges equal to 8Dp. The sphere is 1 mm in diameter. In
28 Chapter 3. The Immersed Boundary Method
u(0) = un + 4tρf· (−∇ · Snu − (3
2Cnu − 1
2Cn−1u )−∇pn)
F(0)k = 0
u(a)k (xk) =
∑xεΩ u(a)(x) · D(x,xk)
F(a+1)k = F
(a)k +
ρf∆t
(ud − u(a)k )
f (a+1)(x) =∑N
k=1 F(a+1)k (xk) · D(x,xk) · ∆Vk
h3
(I− 12
νfh2L)u(a+1) = un+4t
ρf·(−1
2∇·Snu−(3
2Cnu− 1
2Cn−1u )−∇pn+f (a+1))
|u(a+1)k − ud| < ε ‖ a ≥ Max Ite
∇2Φn+1 =ρf4t∇ · u(a+1)
un+1 = u(a+1) − 4tρf∇Φn+1 , pn+1 = pn + Φn+1
a = 0
True
False
a = a + 1
Figure 1: Solution scheme of the IB method to calculate the flow field.
1
Figure 3.2: Solution scheme of the IB method to calculate the flow field.
3.5. Verification 29
T(0) = Tn +4t · (−∇ · SnT − (32CnT − 1
2Cn−1T ))
Q(0)k = 0
T(a)k (xk) =
∑xεΩ T (a)(x) · D(x,xk)
Q(a+1)k = Q(a)
k +Td−T (a)
k
∆t
q(a+1)(x) =∑N
k=1 Q(a)k (xk) · D(x,xk) · ∆Vk
h3
(I − 12
αf
h2L)T (a+1) = T n +4t · (−1
2∇ · SnT − (3
2CnT − 1
2Cn−1T ) + q(a+1))
|T (a+1)k − Td| < ε ‖ a ≥ Max Ite
T n+1 = T (a+1)
a = 0
True
False
a = a + 1
Figure 1: Solution scheme of the IB method to calculate the temperaturefield.
1
Figure 3.3: Solution scheme of the IB method to calculate the temperature field.
30 Chapter 3. The Immersed Boundary Method
0
0.25
0.5
0.75
1
Figure 1: Non-dimensional temperature distribution around a sphere att=0.6 s.
1
Figure 3.4: Non-dimensional temperature distribution of the unbounded fluid aroundthe hot sphere at t=0.6 s.
this simulation a time step of 10−4 s is used whereas the thermal diffusivity is set to
10−6 m2/s. The initial non-dimensional temperature T = (T (r, t)−T∞)/(Ts−T∞) of
sphere and fluid are 1 and 0, respectively. A far field boundary condition (temperature
is assumed to have zero-normal derivative) is imposed on all domain boundaries. Four
grid sizes h = Dp/15, h = Dp/20, h = Dp/25 and h = Dp/30 are used to show grid
convergence.
As time advances, the heat diffuses into the fluid and the temperature continu-
ously increases in the vicinity of the sphere. Fig. 3.4 shows the instantaneous non-
dimensional temperature distribution around the sphere at t = 0.6 s. A comparison
between the analytical solution (Eq. 3.29) and the numerical results in terms of the
radial non-dimensional temperature distribution is presented in Fig. 3.5 at different
times. Table 3.1 reports the analytical (Eq. 3.30) and computed Nusselt numbers at
four different times. Table 3.1 shows that computed Nusselt numbers do not change
noticeably if were obtained from grid sizes of h = Dp/25 and h = Dp/30. The
simulation results are in good agreement with the analytical values.
3.5.2 Forced convection around a stationary sphere
Numerical simulations of an isothermal hot sphere placed in a flowing cold gas were
performed to validate the proposed IB method for forced convection heat transfer.
The calculated mean Nusselt number for a sphere can be compared to the available
empirical correlations. Six different Reynolds numbers (Re = 20, 30, 40, 50, 60 and
100) based on the free-stream gas velocity U∞ and sphere diameter Dp = 1 mm are
considered. The fluid density, viscosity and the Prandtl number Pr are 1 kg/m3,
3.5. Verification 31
1 2 3 4 5 6
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
r/R
Non
-dim
ensi
onal
tem
per
ature
Analytical solutionPresent model
t= 0.1 s
t= 0.3 s
t= 0.6 s
t= 1.0 s
Figure 1: Radial non-dimensional temperature profile, comparison betweennumerical and analytical solutions.
1
Figure 3.5: Radial non-dimensional temperature profile of the unbounded fluid, com-parison between numerical and analytical solutions (h = Dp/25).
Table 3.1: Nusselt number versus time for cooling of the hot sphere in the unboundedfluid
Time Estimated Nusselt No Nusselt No (Eq. 3.30)(s) h = Dp/15 h = Dp/20 h = Dp/25 h = Dp/30 (Analytical value)0.4 3.01 2.97 2.95 2.93 2.890.6 2.83 2.80 2.78 2.77 2.730.8 2.73 2.70 2.68 2.67 2.631.0 2.66 2.63 2.61 2.60 2.56
32 Chapter 3. The Immersed Boundary Method
10−5 kg/(m · s) and 1, respectively. Since the Re number is high enough, natural
convection can be neglected.
We would like to obtain the heat-transfer characteristics for the flow past a sphere
in an infinitely extended fluid. This means that the computational domain needs to
be large enough for boundary effects are negligible. Nijemeisland (2000) proposed
that a domain with 8Dp × 8Dp × 16Dp in size is proper for removing the wall effect
on the velocity and temperature profiles. Guardo et al. (2006) showed that a domain
with size of 4Dp × 4Dp × 16Dp can still be considered as a infinite domain for this
problem. We have taken a size of 8Dp × 8Dp × 15Dp. To study the influence of the
mesh size were presented with using 3 different mesh sizes h = Dp/10, h = Dp/15
and h = Dp/20.
The boundary conditions for this flow are outlined below:
• At the surface of the sphere, no slip (u = 0) and prescribed temperature bound-
ary conditions (T = Ts) are imposed.
• At the inlet, uniform axial velocity (U∞ = uz, ux = uy = 0) and temperature
(T = T∞) of the fluid are imposed.
• At the outlet, the boundary conditions are:
∂u
∂z= 0,
∂T
∂z= 0 (3.31)
• Free slip boundary condition is used for other boundaries:
∂u
∂y=∂u
∂x= 0,
∂T
∂x=∂T
∂y= 0 (3.32)
The inlet and outlet boundaries are located at z = −2Dp and 13Dp, respectively,
where the center of the sphere is at the origin. Fig. 3.6 shows the distribution of
the non-dimensional gas temperature, T = (T (x, y, z)−T∞)/(Ts−T∞), accompanied
with the velocity vector around the sphere at the Z−Y plane. The Nusselt number is
estimated by evaluating the convective heat transfer coefficient hf around the surface
of the sphere,
hf =QT
(T∞ − Ts)Ap, (3.33)
where QT and Ap are total heat flux and surface area of the sphere.
Fig 3.7 reports the calculated Nusselt number with the empirical results of Ranz
and Marshall (1952), Whitaker (1972) and Feng and Michaelides (2000) for a spherical
particle, respectively:
Nu = 2.0 + 0.6 Re0.5Pr0.33 for 10 < Re < 104,Pr > 0.7. (3.34)
3.5. Verification 33
0
0.25
0.5
0.75
1
Figure 1: Distribution of non-dimensional gas temperature around a heatedsphere together with velocity field at Re=40 and Pr=1.
1
Figure 3.6: Distribution of non-dimensional gas temperature around a heated spheretogether with velocity field at Re=40 and Pr=1.
Nu = 2.0 + (0.4 Re0.5 + 0.06 Re2/3) Pr0.4
for 3.5 < Re < 7.6 × 104, 0.7 < Pr < 380. (3.35)
Nu = 0.992 + Pe1/3 + 0.1 Re1/3Pe1/3
Pe = Re× Pr, for 0.1 < Re < 4000, 0.2 < Pe < 2000. (3.36)
The sensitivity of the numerical solutions to grid sizes can be examined from Fig.
3.7. There are almost no differences between the results from grid sizes h = Dp/15 and
h = Dp/20 when the Reynolds number is lower than 50 (around 1%). This difference
is around 3% when the Reynolds number is 100. The calculated Nusselt numbers are
in reasonable agreement with the results obtained from empirical correlations.
3.5.3 Effect of blockage on flow and heat transfer from an in-line
array of three spheres
In this section, we perform computations for flow over an array of three spheres
as shown schematically in Fig. 3.8 together with the details of the computational
domain. In order to investigate the combined effects of blockage and of sphere-sphere
interactions, the flows past an in-line array of three spheres have been studied for
three different Reynolds numbers (Re=1, 10, 50) and for two values of the center-to-
center spacing between the spheres, namely, s = 2Dp and 4Dp. The same values were
34 Chapter 3. The Immersed Boundary Method
10 20 30 40 50 60 70 80 90 1001
2
3
4
5
6
7
8
9
10
Re
NusseltNo.
Ranz and Marshall (1952)Whitaker (1972)Feng and Michaelides (2000)Present simulations (h = Dp/20)Present simulations (h = Dp/15)Present simulations (h = Dp/10)
1
Figure 3.7: Nusselt number versus Reynolds number for forced convection over asphere.
Sphere
s s
DtDpz
y
T0
U0
Figure 1: Schematic of flow over a row of three spheres.
1
Figure 3.8: Schematic of flow over a row of three spheres
used by Ramachandran et al. (1989) and Maheshwari et al. (2006) in simulations of
heat transfer for unconfined in-line arrays of three spheres.
The parameters used in our simulations study are provided in table 3.2. The den-
sity and viscosity of the gas are 1 kg/m3 and 10−5 kg/(ms), respectively. According
to the results of mesh independency tests for forced convection around a sphere (Fig.
3.7), the numerical results are mesh independent (when Re ≤ 50) if the grid size h
is set to Dp/15. The boundary conditions for these simulations are the same as the
boundary conditions used in the simulation of forced convection around a stationary
sphere in the previous section. As in all presented simulations, the thermo-physical
properties are taken to be temperature independent.
3.5. Verification 35
Table 3.2: Parameters used in simulations of heat transfer from an in-line array ofthree spheres.
s/Dp Nx×Ny ×Nz Dp(m) Dt/Dp r1/Dp r2/Dp Grid Size (m) Pr2 150× 150× 195 10−4 10 4.5 4.5 6.666× 10−6 0.744 150× 150× 255 10−4 10 4.5 4.5 6.666× 10−6 0.74
0
0.25
0.5
0.75
1
Figure 1: Distribution of non-dimensional gas temperature around heatedspheres together with the velocity field at Re = 5 and s/Dp =2.
1
Figure 3.9: Distribution of non-dimensional gas temperature around heated spherestogether with the velocity field at Re = 10 and s/Dp =2.
The distribution of the non-dimensional gas temperature as well as the velocity
field around the spheres in the z − y plane are shown in Fig. 3.6. The present
results are compared with results of Ramachandran et al. (1989) and Maheshwari
et al. (2006) for air (Pr=0.74) in Table 3.3 for the two values of the sphere-to-sphere
separation. According to the numerical results, higher Reynolds numbers will increase
the heat exchange rate between gas and particles. Moreover, a larger sphere-to-sphere
separation ratio leads to the higher heat transfer rates.
The overall agreement is acceptable, but around 5-7 % deviation is observed for the
first and third spheres. A possible reason is the size of the entrance and exit domains
(e.g. r1 and r2) on solutions. These sizes are not given by Maheshwari et al. (2006)
and we found that changing them influences the results for the first and third spheres
significantly (up to 15 % difference). An additional reason for the deviation might be
the presence of thin boundary layers, especially for the first sphere, and insufficient
grid resolution to resolve these. The accurate simulation of this problem will result in
high computational costs since a uniform grid is used and upon refinement a refined
grid must be used for the whole domain. Hence, in this study we do not consider
36 Chapter 3. The Immersed Boundary Method
Table 3.3: Comparison between the present results of Nusselt number for an in-linearray of three spheres and that of Ramachandran et al. (1989) and Maheshwari et al.(2006) for air (Pr=0.74).
Re S/Dp Present Simulation Ramachandran et al. Maheshwari et al.1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd
1 2 2.09 1.58 1.62 2.12 1.81 1.63 2.03 1.83 1.631 4 2.31 1.96 1.82 2.17 2.03 1.63 2.20 1.94 1.6410 2 3.45 2.40 2.21 3.37 2.32 2.03 3.32 2.34 2.0510 4 3.51 2.83 2.62 3.28 2.79 2.49 3.33 2.72 2.5350 2 5.72 3.55 3.19 5.50 3.39 2.98 5.42 3.44 3.0850 4 5.80 4.21 3.81 5.40 4.18 3.60 5.40 4.11 3.52
using a larger domain or a more refined grid for this problem.
3.6 Conclusions
In the present chapter, the IB method with a finite difference scheme proposed by
Uhlmann is extended to heat transfer applications. In this method, a direct heat
source term is introduced to enforce the temperature Dirichlet boundary condition at
an IB and is estimated in an iterative manner. Thus it inherits the advantages of the
IB method as well in heat transfer applications and can be used efficiently in non-
isothermal complex geometry problems. For our computational scheme, the implicit
second order fractional step method and the discrete element particle approach are
employed to simulate the behavior of fluid and solid phases, respectively.
The accuracy of proposed numerical method was validated by comparing results
obtained for several heat transfer problems involving single and multi-particle sys-
tems with well-known correlations. First, the heat conduction problem around an
isothermal hot sphere immersed in an infinite stationary fluid was simulated. The
computed temperature profile and associated Nusselt number agreed well with ana-
lytical solutions. Then the forced convection around a single sphere and an in-line
array of three spheres for several Reynolds numbers were simulated. The estimated
Nusselt numbers were compared with the data published in literature.
4
CH
AP
TE
R
DNS of random arrays of monodisperse
spheres
Abstract
Heat transfer characteristics in particulate flow are of the utmost importance for the
design of industrial equipment such as packed and fluidized beds. Well-known heat
transfer correlations for such systems can differ significantly from each other. DNS
of heat transfer in gas-solid flow can improve our insight about complex transport
phenomena in such particulate systems. In addition, the heat transfer can be fully
quantified for these well-defined systems. In this study, gas-particle HTCs obtained
from DNS are compared with well-established empirical correlations over a wide range
of solids volume fractions at Pr=1 and Reynolds numbers ranging between 1 and 100.
Furthermore, a detailed analysis reveals that the particle-fluid heat exchange is af-
fected strongly by heterogeneity of the bed. The difference between particle and the bed
HTCs is quantified. This deviation is by construction ignored in coarse scale methods
such as the Discrete Particle Model. The individual particle HTC in a statistically
homogeneous array can differ up to 60% from the HTC of the bed. Finally we show
that by defining the HTCs based on the local Reynolds number and fluid temperature,
this deviation decreases.
37
38 Chapter 4. DNS of random arrays of monodisperse spheres
4.1 Introduction
DNS, where all relevant scales are resolved, can be used to generate drag and heat-
transfer correlations. These correlations can be used in a DPM, where the gas phase
is not fully resolved. The information on the behavior of the particle phase that is
resolved in DPM can be used in a TFM where this phase is modeled as a pseudo
fluid.
In the progressively coarser methods it is attempted to simulate the systems at
larger time and length scales within a reasonable amount of computational time,
but without loosing too much of the important phenomena. In the coarse-grained
methods, only the large length scales are resolved. The influence of smaller scale
phenomena on these large scales are approximated with the help of closure relations.
In other words, not all details of the flow are predicted. Therefore in the coarse scale
approach, the closure models have significant effect on macroscopic behavior of the
system.
In current practice many closure relations used in e.g. DPM are not obtained
from DNS but are empirical engineering correlations. This type of correlation is of
a macroscopic nature. In DPM these correlations are applied for each particle indi-
vidually using ‘local’ particle-based Reynolds numbers and solids volume fractions.
These local quantities, however, are usually defined on the grids used for solving the
transport equations of the gas and interpolated to the particle positions. Because the
gas is solved on a coarse grid compared to the particle there is implicitly a type of
averaging over the grid size. In other words, when computing momentum and heat
transport in DPM a mean drag and HTC are used for each particle instead of the
‘real’ values. This raises questions that can be answered by DNS. For example: How
accurate is an empirical HTC for a well defined particle arrangement, or: How big
is the loss of accuracy due to the fact that local heterogeneity is not fully accounted
for?
Although fully resolved simulation of a large multiphase system is not feasible, the
DNS of a small system can increase our insight into transport phenomena in complex
multiphase systems. In the DNS approach, the fluid and solid phases are treated
by considering the Navier-Stokes equations and the Newtonian equations of motion,
respectively. The mutual interactions between the phases are obtained by enforcing
the appropriate boundary conditions at the surface of the particle (e.g. no-slip and
Dirichlet boundary conditions for momentum and heat transfer, respectively) and
therefore no empirical correlations are required. Consequently, the drag and HTCs
can be determined from DNS results. With this clear advantage of DNS the accuracy
of well-known correlations can be assessed with DNS results obtained from well-
defined systems.
Drag in particulate systems was previously studied with the help of DNS (Koch
et al. (1997), Beetstra et al. (2007), Tang et al. (2014)). They performed extensive
4.2. Simulation details 39
simulations over a wide range of Reynolds numbers and solids volume fractions. Sub-
sequently correlations were proposed to evaluate the drag on the basis of simulation
data. Recently few studies have been undertaken to characterize the HTC in packed
and fluidized beds with the help of DNS (Deen et al. (2012), Tenneti et al. (2013),
Tavassoli et al. (2013)). In all of these studies significant deviations (up to 30%) be-
tween numerical and experimental results are observed. This deviation can be caused
by the uncertainty in the experimental data and in the numerical model.
Kriebitzsch et al. (2013) employed DNS to study the flow through a static random
array of particles. Kriebitzsch et al. investigated the deviation between the true drag
force (obtained from DNS) and the drag that acts on individual particles in packed
and fluidized beds using DPM. They found that on average the mean DPM fluid-solid
drag force is significantly smaller than the true value (around 20-30 %) in packed and
fluidized beds. In fact, this deviation originates from the local heterogeneity of the
system. It must be noted that this result was obtained for a statistically homogeneous
systems. For heterogeneous systems, the deviation of the true drag and the one
estimated from correlations will likely be larger. As far as the authors know no effort
has been made to investigate the effect of the local heterogeneity on HTCs in packed
and fluidized beds.
All of these facts motivate us to investigate and compare experimental and thermal
models for packed and fluidized beds in more detail with the help of DNS. The results
of this study enable us to improve the prediction of the numerical simulation of heat
transfer in particulate systems.
In this study, we employ the IBM to study non-isothermal flows through stationary
arrays of spherical particles. The computed mean HTC of the bed will be compared
with well-known heat transfer correlations for packed and fluidized beds. Next, we
will characterize the deviation of particle HTCs from the average value in terms
of operation conditions. This chapter is organized as follows. First, the HTCs of
a fixed bed of particles are compared with well-known correlations. The effect of
heterogeneity on the particle heat flux is investigated as well. Last, the summary and
conclusion are given in the final section.
4.2 Simulation details
The physical model explained in section 2.6.1 is used to characterize the heat transfer
in a fixed random array of spherical particles.
The grid independency of solutions has been investigated by performing simula-
tions using maximum 4 different grid sizes equal to Dp/20, Dp/30, Dp/40 and Dp/50
for each case. The total heat exchange rates of a bed obtained by using different mesh
sizes were compared with each other. When the results show the desired convergence
(i.e. when the deviation is lower than 2%), the results were considered to be “mesh-
independent”. In general, the proper mesh size is function of Reynolds number and
40 Chapter 4. DNS of random arrays of monodisperse spheres
φ. Table 4.1 reports the employed mesh size for each case in our simulations.
Table 4.1: The ratio of particle diameter to the grid size in order to obtain meshindependent results.
Re, φ 0.1 0.2 0.3 0.4 0.5 0.6
10 30 30 40 40 40 4030 30 30 40 40 40 4050 30 30 40 50 50 5070 30 30 40 50 50 50100 30 30 40 50 50 50
4.3 Results
4.3.1 The mean heat transfer coefficient in a random array of
particles
According to calculated velocity and temperature distributions, the HTC of the bed
can be obtained from Eq. (2.2). Due to the variation of the fluid-particle interface in
the cross-sectional planes along the flow direction, a variation of the HTC is observed
in the bed. A better estimation of the mean HTC can be obtained from the average
value of HTCs over different configurations. Fig. 4.1 shows the distribution of Nusselt
number along the flow direction in the bed. This result is obtained from averaging
over 5 independent configurations when the Reynolds number and φ are 50 and 0.4,
respectively. We can estimate the mean Nusselt number by averaging Nu(x) along
the flow direction.
As previously mentioned, the inlet and outlet sections affect the value of the
average HTC. If a bed is long enough there will be a region in the middle of the
bed where the HTC is constant. Near the entrance of the bed, x = 0, and near
the exit, x = X, deviations from this constant value are expected due to inlet and
outlet effects. These effects can be investigated and characterized with the help of
numerical simulations. At the inlet and outlet of the bed in Fig. 4.1, we see a
significant change in Nu(x) in comparison to the value in the central section of the
bed. It is difficult to characterize this deviation in terms of operating conditions.
However, we have observed up to 10% deviation in our simulations. These results
reveal one of the reasons of the deviation between experimental and numerical results
(when the periodic boundary condition is used in streamwise direction e.g. Tenneti
et al. (2013)).
Our DNS indicates that Nu(x) approaches to a constant value after approximately
2Dp from the entrance of the bed when the solids volume fraction ranges from 0.1
4.3. Results 41
to 0.6. The exit region is smaller than 1Dp in all cases. Since the physical domains
used in this study are larger than 3Dp in streamwise direction, we can conclude that
bed is large enough and consequently the correct mean HTC can be obtained from
the numerical simulations. For the estimation of the mean bulk Nusselt number in
this study, the inlet and outlet regions were excluded.
Table 1: The ratio of particle diameter to the grid size in order to obtain mesh independent results.
Re, φ 0.1 0.2 0.3 0.4 0.5 0.6
10 30 30 40 40 40 4030 30 30 40 40 40 4050 30 30 40 50 50 5070 30 30 40 50 50 50100 30 30 40 50 50 50
5 Results
5.1 The mean heat transfer coefficient in a random array of particles
According to calculated velocity and temperature distributions, the HTC of the bed can be obtained fromEq. (2). Due to the variation of fluid-particle interface in the cross-sectional planes along the flow direction, avariation of the HTC is observed in the bed. A better estimation of HTC (mean value) can be obtained fromthe average value of HTC over different configurations. Fig. 3 shows the distribution of Nusselt number alongthe flow direction in the bed. This result is obtained from averaging over 5 independent configurations when theReynolds number and φ are 50 and 0.4, respectively. We can estimate the mean Nusselt number by averagingNu(x) along the flow direction.
As previously mentioned, the inlet and outlet sections affect the value of the average HTC. If a bed islong enough there will be a region in the middle of the bed where the HTC is constant. Near the entrance ofthe bed, x = 0, and near the exit, x = X, deviations from this constant value are expected due to inlet andoutlet effects. These effects can be investigated and characterized with the help of numerical simulation. Atthe inlet and outlet of the bed in Fig. 3, we see a significant change in Nu(x) in comparison to the value inthe central section of the bed. It is difficult to characterize this deviation in terms of operating conditions.However, we have observed up to 10% deviation in our simulations. These results reveal one of the reasonsof the deviation between experimental and numerical results (when the periodic boundary condition is used instreamwise direction e.g. Tenneti et. al. (2013)).
Our DNS indicates that Nu(x) approaches to a constant value after around 2 dp (entrance region) far fromthe entrance of the bed when the solids volume fraction ranges from 0.1 to 0.6. The exit region is smaller than1 dp in all cases. Since the physical domains used in this study are larger than 3 dp in streamwise direction,we can conclude that bed is large enough and consequently the correct mean HTC can be obtained from thenumerical simulations. For the estimation of the mean bulk Nusselt number in this study, the inlet and outletregions were consequently excluded.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
8
10
12
14
Inlet
Outlet
x/L
Nu(x)
(a)
Figure 3: The local Nusselt number Nu(x) along the flow direction in a bed for Re=50 and φ = 0.4.
7
Figure 4.1: The local Nusselt number Nu(x) along the flow direction in a bed forRe=50 and φ = 0.4.
In Fig. 4.2 the Nusselt numbers obtained from DNS are compared with the
empirical correlation proposed by Gunn (Eq. 2.9). The general trend of the DNS
results follows the Gunn correlation. The agreement between the computed Nusselt
number and the results obtained from the Gunn correlation is quite reasonable when
φ is lower than 0.3. However, a significant deviation is observed when φ is higher than
0.3. This large deviation is observed for the whole range of Reynolds numbers studied.
In order to examine the accuracy of the Gunn correlation for dense systems, e.g.
packed beds, we compare our results with another well-known correlation proposed
by Wakao (Eq. 2.8) for dense packed bed (φ ≈ 0.6).
Fig. 4.3 shows the Nusselt numbers for a packed bed (φ = 0.6) obtained by the
Gunn and Wakao correlations together with the numerical simulation results. It can
be seen that the numerical results agree well with Wakao predictions. Therefore, the
42 Chapter 4. DNS of random arrays of monodisperse spheres
Gunn correlation is not proper for dense systems (φ > 0.3) although it predicts the
Nusselt number well for dilute systems when the Reynolds number is lower than 100.
Based on our simulation data, we refit the Gunn correlation and extract the
following relationship for the convective HTC in packed and fluidized beds (Fig. 4.2):
NuRefit = (7.0−10ε+5ε2)(1+0.1Re0.2Pr1/3)+(1.33−2.19ε+1.15ε2)Re0.7Pr1/3
for 0 < Re < 102, and 0.35 < ε = (1 − φ) < 1 (4.1)
An estimate of the HTC can also be obtained by analyzing the inflow and outflow
temperatures using Eq. (2.4). This estimate discards dispersion effects and it is
expected that hno−disp > 〈h(x)〉. Table 4.2 reports the percentage error between these
two HTCs. Our results reveal that the effect of dispersion increases with decreasing
Reynolds number and increasing φ. The highest deviation is around 11.5% when
Re= 10 and φ = 0.6. This deviation is lower than 4% when Reynolds number exceeds
70.
Table 4.2: The percentage error of heat transfer coefficient when the axial dispersioneffect is neglected.
Re, φ 0.1 0.2 0.3 0.4 0.5 0.6
10 2.35 4.23 4.55 7.39 9.05 11.6330 2.45 3.49 3.93 5.37 4.83 4.650 2.71 2.9 3.68 5.2 4.92 4.770 2.7 2.71 3.39 4.31 4.43 3.92100 1.3 2.85 3.40 4.28 4.12 4.01
4.3. Results 43
00.1
0.2
0.30.4
0.5
0.6
5
10 15 20 25
φ
Nusselt No.
Sim
ulation
(Re=
10)
Sim
ulation
(Re=
30)
Sim
ulation
(Re=
50)
Sim
ulation
(Re=
70)
Sim
ulation
(Re=
100)
Gu
nn
(Re=
10)
Eq.(4.1)
(Re=
10)
Gu
nn
(Re=
30)
Eq.(4.1)
(Re=
30)
Gu
nn
(Re=
50)
Eq.(4.1)
(Re=
50)
Gu
nn
(Re=
70)
Eq.(4.1)
(Re=
70)
Gu
nn
(Re=
100)
Eq.(4.1)
(Re=
100)
Fig
ure
4:
Th
em
eanN
usselt
nu
mb
erin
ran
dom
arrays
ofsp
here
obtain
edfrom
Gu
nn
correlationan
dnu
merical
simu
lation
s.
InF
ig.4
the
Nu
sseltnu
mb
ersob
tained
fromD
NS
arecom
pared
with
the
emp
iricalcorrelation
prop
osedby
Gu
nn
:
Nu
Gunn
=(7−
10ε
+5ε
2)(1+
0.7R
e0.2P
r1/3)
+(1.33−
2.4ε
+1.2ε
2)Re0.7P
r1/3
for0<
Re<
105,
and
0.35
<ε<
1(15)
wh
ereε
isvoid
age
ofth
eb
ed(ε
=1−
φ).
Th
egen
eral
trend
ofth
eD
NS
results
follows
the
Gu
nn
correlation.
Th
eagreem
ent
betw
eenth
ecom
pu
tedN
usselt
nu
mb
eran
dth
eresu
ltsob
tained
fromth
eG
un
ncorrelation
isqu
itereason
able
wh
enφ
islow
erth
an0.3.
How
ever,
asign
ifican
td
eviation
isob
served
wh
enφ
ish
igher
than
0.3.T
his
larged
eviatio
nis
observed
forth
ew
hole
ran
geof
Rey
nold
snu
mb
ersstu
died
.In
order
toex
amin
eth
eaccu
racyof
the
Gu
nn
correlationfor
den
sesy
stems,
e.g.p
ackedb
eds,
we
comp
areou
rresu
ltsw
ithan
other
well-k
now
ncorrelation
prop
osedby
Wakao
for
den
sep
acked
bed
(φ≈
0.6).
Nu
Wakao
=2
+1.1R
e0.6P
r1/3
(16)
Fig
.5
show
sth
eN
usselt
nu
mb
ersfo
ra
packed
bed
(φ=
0.6)ob
tained
by
Gu
nn
and
Waka
ocorrelation
stog
ether
with
the
nu
merical
simu
lationresu
lts.It
canb
eseen
that
the
nu
merical
results
agreew
ellw
ithW
akaop
redictio
ns.
Th
erefore,th
eG
un
ncorrela
tion
isn
otp
roper
ford
ense
system
s(φ>
0.3)
althou
ghit
pred
ictsth
eN
usselt
nu
mb
erw
ellfo
rd
ilute
system
sw
hen
the
Rey
nold
snu
mb
eris
lower
than
100.B
ased
onou
rsim
ulation
data
,w
erefi
tth
eG
un
ncorrelation
and
extract
the
followin
grelation
ship
forth
eco
nvective
HT
Cin
packed
and
flu
idized
bed
s(F
ig.4):
Nu
Refit
=(7.0−
10ε+
5ε
2)(1+
0.1Re
0.2P
r1/3)
+(1.33−
2.19ε
+1.15ε
2)Re0.7P
r1/3
for0<
Re<
102,
and
0.35
<ε<
1(17)
An
estimate
ofth
eH
TC
canalso
be
obtain
edby
analy
zing
the
infl
owan
dou
tflow
temp
eratures
usin
gE
q.
(4).
Th
isestim
ated
iscards
disp
ersion
effects
and
itis
exp
ectedth
ath
no−
disp
>〈h
(x)〉.
Tab
le2
reports
8
Figure 4.2: The mean Nusselt number in random arrays of sphere obtained from theGunn correlation and numerical simulations.
44 Chapter 4. DNS of random arrays of monodisperse spheres
010
2030
4050
6070
8090
1000 5 10 15 20 25
Rey
nold
snu
mb
er
Nusselt No.
Sim
ula
tionG
un
ncorrelation
Wakao
correlation
Figu
re5:
Th
em
eanN
usselt
num
ber
ina
packed
bed
obtain
edfrom
Gu
nn
and
Wakao
correlationan
dnu
merica
lsim
ulation
s(φ
=0.6).
the
percen
tage
errorb
etween
these
two
HT
C’s.
Ou
rresu
ltsreveal
that
the
effect
ofd
ispersion
increases
with
decrea
sing
Rey
nold
snu
mb
eran
din
creasingφ
.T
he
high
estd
eviation
isarou
nd
11.5%
wh
enR
e=10
andφ
=0.6
.T
his
dev
iation
islow
erth
an4%
wh
enR
eyn
olds
nu
mb
erex
ceeds
70.
Tab
le2:
Th
ep
ercenta
ge
error
ofh
eattran
sferco
efficien
tw
hen
the
axial
disp
ersion
effect
isn
eglected.
Re,φ
0.10.2
0.30.4
0.50.6
102.35
4.234.55
7.399.05
11.6330
2.453.49
3.935.37
4.834.6
502.71
2.93.68
5.24.92
4.770
2.72.71
3.394.31
4.433.92
1001.3
2.853.40
4.284.12
4.01
5.2
Asse
ssment
of
particle
heat
tran
sfer
coeffi
cient
ina
ran
dom
arra
y
As
discu
ssedin
the
prev
iou
ssectio
n,
the
heat
exch
ange
rateth
rough
packed
and
flu
idized
bed
sis
characterized
by
the
mean
Nu
sseltnu
mb
eran
din
fact
this
param
eterrep
resents
the
averageof
allin
div
idu
alp
article-flu
idN
usselt
nu
mb
ersN
up .
Sin
ceN
up
isu
sedfor
characterizin
gth
eparticle-fl
uid
heat
flu
xQ
pin
coarse-grain
edap
proa
ches,
the
intrigu
ing
qu
estionis
how
mu
chth
eN
up ,
onaverage,
dev
iatesfrom
the
mean
value
ina
bed
.A
san
exam
ple,
Fig.
6(a)sh
ows
valu
esofQ
pof
ind
ivid
ual
particles
along
the
length
ofth
ep
acked
section
wh
enRe
=50
andφ
=0.4.
As
exp
ected,Q
pis
high
atth
ein
letof
the
bed
and
decreases
along
the
length
of
the
bed
.F
ig.
6(a
)clearly
show
sth
atth
ep
articles
with
atth
esam
ep
ositionalon
gth
efl
owd
irection(x
directio
n)
canex
perien
cesign
ifica
nt
diff
erent
hea
tfl
uxes.
Sin
ceQ
pis
defi
ned
asth
ep
rod
uct
ofhp
and
the
therm
ald
rivin
gfo
rce∆T
,w
ecan
conclu
de
that
atleast
eitherhp
or∆T
isn
otcon
stant
ata
crosssection
perp
end
icular
toth
efl
ow.
Th
isis
incon
trastto
the
assum
ption
sem
ployed
insim
plifi
edm
od
elssu
chas
Eq.
(3)an
dE
q.
(5)w
here
both
∆T
and
HT
Con
lych
ange
alon
gth
efl
owd
irection.
Fig.
6(b
)sh
ows
the
variation
ofp
articleN
usselt
nu
mb
er(N
up
=hp d
p /k)
aroun
dth
em
eanN
usselt
nu
mb
erN
ub
=〈h
(x)〉d
p /kof
the
system
.A
signifi
cant
flu
ctuation
ofN
up
isob
served
.In
order
toqu
antify
this
9
Figure 4.3: The mean Nusselt number in a packed bed obtained from the Gunn andWakao correlation and numerical simulations (φ = 0.6).
4.4. Influence of micro-structure on particle-fluid heat transfer rate 45
4.3.2 Assessment of particle heat transfer coefficient in a random
array
As discussed in the previous section, the heat exchange rate through packed and
fluidized beds is characterized by the mean Nusselt number and in fact this parameter
represents the average of all individual particle-fluid Nusselt numbers Nup. Since Nupis used for characterizing the particle-fluid heat transfer rates Qp in coarse-grained
approaches, the intriguing question is how much the Nup, on average, deviates from
the mean value in a bed.
As an example, Fig. 4.4a shows values of Qp of individual particles along the
length of the packed section when Re= 50 and φ = 0.4. As expected, Qp is high
at the inlet of the bed and decreases along the length of the bed. Fig. 4.4a clearly
shows that the particles at the same position along the flow direction (x direction)
can experience significant different heat transfer rates. Since Qp is defined as the
product of hp, Ap and the thermal driving force ∆T , we can conclude that at least
either hp or ∆T is not constant at a cross section perpendicular to the flow. This
is in contrast to the assumptions employed in simplified models such as Eq. (2.3)
and Eq. (2.5) where both ∆T and HTC are assumed to only change along the flow
direction.
Fig. 4.4b shows the variation of particle Nusselt number (Nup = hpDp/k) around
the mean Nusselt number Nu = 〈h(x)〉Dp/k of the system. A significant fluctuation
of Nup is observed. In order to quantify this fluctuation, we calculate the relative
standard deviation S can be computed as
S =1
〈Nup〉
√∑Np
i=1(Nuip − 〈Nup〉)2
Np − 1, with 〈Nup〉 =
1
Np
Np∑
i=1
Nuip. (4.2)
Fig. 4.5 reports the values of S as function of φ and the Reynolds number. For
providing these results, 5 independent simulations with 54 particles were performed
for each solids volume fraction and Reynolds number. In order to obtain a better
estimate of S, the particles that are influenced by entrance effects are excluded from
the analysis. The results show that S increases with increasing φ. S is as high as
60% when the Reynolds number and φ are 30 and 0.6, respectively.
In the next sections the reasons of this significant deviation are investigated in
more detail and it is tried to relate the Nup to the local micro-structural information.
4.4 Influence of micro-structure on particle-fluid
heat transfer rate
The previous results indicate that the local micro-structure around each particle has
a strong effect on Qp. Therefore a better estimate of the individual fluid-particle
thermal interaction is required. Accordingly the impact of the local micro-structure
46 Chapter 4. DNS of random arrays of monodisperse spheres
(a)
(b)
Figure 4.4: a) The particle-fluid heat transfer rate b) The particle-fluid heat transfercoefficient along the bed obtained by DNS (Re = 50 and φ = 0.4).
4.4. Influence of micro-structure on particle-fluid heat transfer rate 47
0.1 0.2 0.3 0.4 0.5 0.60.2
0.4
0.6
0.8
φ
S
Re=30Re=50Re=70Re=100
Figure 7: Mean relative deviation S (Eq. (18)) of the particle Nusselt number and the mean Nusselt number ofthe bed Nub for different Reynolds number.
Figs. 8(a) and 8(b) show the temperature field and associated magnitudes of Nup and Qp in a cross sectionalong the flow direction when Re = 70 and φ = 0.4. As observed in Fig. 8(a), the fluid temperature increasesalong the flow direction due to heat exchange with hot particles. Therefore, the first particles upstream of theflowing fluid feel a higher thermal driving force (all particles have the same constant temperature) and havea higher heat exchange rate with the fluid in comparison to downstream particles. Fig. 8(a) shows that theeffect of upstream particles on the thermal driving force for all downstream particles on a plane is not the same.Consequently, the particles present in the same plane experience a different thermal driving force as well asdifferent heat flux.
The thermal driving force required for the calculation of Nup is obtained as the difference of particle tem-perature and an average fluid temperature around the particle. In our numerical results for Nup the cup-mixingtemperature of fluid on the plane perpendicular to the flow direction is used. Fig. 8(b), shows a fluctuation ofNup over the mean value in the bed especially for the particles at the end of the bed. It seems this discrepancyoriginates from considering the too large region around each particle used to define the average fluid temperature(all particles in the spanwise plane feel the same thermal driving force). A particle Nusselt number based onmore localized average temperatures might show less variation. We will come back to this issue in the followingsections.
It is known that there is a direct relationship between the velocity magnitude and the forced convectionheat transfer. Due to the heterogeneous local φ distribution, originating from the random particle arrangement,the local velocity magnitude is nonuniform in the bed. The descriptions of local velocity and φ (or the localReynolds number) in a bed are not so straightforward. In order to assess the relation between Qp and the localReynolds number, we describe the local velocity as follows. Around each particle a centered cube with a volumeπD3/6φ. These volumes, by definition, add up to the total volume. Next, the fluid entering the upstream faceof the cube is used to compute a local superficial velocity. This local superficial velocity is used to define alocal Reynolds number based. Fig. 8(c) shows the fluid velocity field through the bed with the local Reynoldsnumber. Comparing Fig. 8(b) and 8(c) clearly shows that high Nup is correlated to a high local Reynoldsnumber.
Fig. 8 shows that significant variations of Nup exist in a statistically homogeneous assembly of monodispersespheres. These results also show that the variation is correlated to local properties. In the next sections wepropose a description for the local Reynolds number, thermal driving force and φ.
11
Figure 4.5: Mean relative deviation S (Eq. (5.7)) of the particle Nusselt number andthe mean Nusselt number of the bed Nub for different Reynolds number.
around each individual particle must be considered. It is known that the local fluid
velocity and the thermal driving force have a direct effect on Qp. On the other hand,
the local φ affects Qp indirectly through the local flow field. In this section we try to
gain a good qualitative understanding of the parameters that influence Qp.
Figs. 4.6a and 4.6b show the temperature field and associated magnitudes of Qpand Nup in a cross section along the flow direction when Re = 70 and φ = 0.4. As
observed in Fig. 4.6a, the fluid temperature increases along the flow direction due
to heat exchange with hot particles. Therefore, the first particles upstream of the
flowing fluid feel a higher thermal driving force (all particles have the same constant
temperature) and have a higher heat exchange rate with the fluid in comparison
to downstream particles. Fig. 4.6a shows that the effect of upstream particles on
the thermal driving force for all downstream particles on a plane is not the same.
Consequently, the particles present in the same plane experience a different thermal
driving force as well as different heat flux.
The thermal driving force required for the calculation of Nup is obtained as the
difference of particle temperature and an average fluid temperature around the par-
ticle. In our numerical results for Nup the cup-mixing fluid temperature on the plane
perpendicular to the flow direction is used. Fig. 4.6b, shows a fluctuation of Nup over
the mean value in the bed especially for the particles at the end of the bed. It seems
that this discrepancy originates from considering the too large region around each
particle used to define the average fluid temperature (all particles in the spanwise
48 Chapter 4. DNS of random arrays of monodisperse spheres
plane feel the same thermal driving force). A particle Nusselt number based on more
localized average temperatures might show less variation. We will come back to this
issue in the following sections.
It is known that there is a direct relationship between the velocity magnitude and
the forced convection heat transfer. Due to the heterogeneous local φ distribution,
originating from the random particle arrangement, the local velocity magnitude is
non-uniform in the bed. The descriptions of local velocity and φ (or the local Reynolds
number) in a bed are not so straightforward. In order to assess the relation between
Qp and the local Reynolds number, we describe the local velocity as follows. Around
each particle a centered cube with a volume πD3p/6φ is defined. These volumes, by
definition, add up to the total volume. Next, the fluid entering the upstream face of
the cube is used to compute a local superficial velocity. This local superficial velocity
is used to define a local Reynolds number. Fig. 4.6c shows the fluid velocity field
through the bed with the local Reynolds number. Comparing Fig. 4.6b and 4.6c
clearly shows that a high Nup is correlated to a high local Reynolds number.
Fig. 4.6 shows that significant variations of Nup exist in a statistically homoge-
neous assembly of monodisperse spheres. These results also show that the variation
is correlated to local properties. In the next sections we propose a description for the
local thermal driving force and φ.
4.4. Influence of micro-structure on particle-fluid heat transfer rate 49
275
290
305
320
(a)
275
290
305
320
(b)
0
2
4
6
(c)
Figure 8: a) Fluid temperature in a X-Z plane (y=0.3Y) with particle heat flux (W/m2). b) Fluid temperaturein a X-Z plane (y=0.3Y) with particle Nusselt number. c) Fluid velocity (|~u(x, y, z)|/U) in a X-Z plane (y=0.3Y)with particle Reynolds number for global Reynolds number = 70 and φ = 0.4 .
6.1 Local HTC
Up to now we computed HTC’s using the cup-averaged temperature computed over the full spanwise directionas given by Eq. (1). This averaging includes parts of space that are far away from the particle at hand. We alsosaw that the particle HTC’s, hp, are variable. Part of this variability is due to the fact that we do not considera local thermal driving force. In this section it will be demonstrated that the use of a locally defined thermaldriving force indeed reduces the variability of computed particle HTC’s.
Consider the particle A inside the non-homogeneous porous system in Fig. 9. In this picture a local controlvolume is drawn. This is the control volume used to compute the local gas temperature. Our simulations showthat two factors have a strong impact on the distribution of a local Nup,loc. First, the size of the control volumearound the particle and, second, the averaging method (i.e. arithmetic mean or cup-mixing temperature).According to these facts, different potential definitions of thermal driving force were analyzed.
We have considered several sizes of control volumes and finally settled on a cubic one centered at the particlewith sides of length 3 dp. For the local gas temperature different definitions can be imagined, namely: simplyaveraging over the volume, using distance dependent weight functions, or using some type of cup-averaging.We found that this last choice gave the best results. Here a local temperature is defined as the cup-averagedtemperature in the cube centered around the particle (indicated as Vf (~xp))
Tf,loc(~xp) =
∫∫∫Vf (~xp)
U(~x)T (~x)dVf∫∫∫Vf (~xp)
U(~x)dVf. (19)
Using this local temperature, the thermal driving force is the temperature difference ∆Tloc = Ts−Tf,loc andthe local particle HTC can be computed as,
hp,loc =Qp
Ap ∆Tloc. (20)
12
(a)
275
290
305
320
(a)
275
290
305
320
(b)
0
2
4
6
(c)
Figure 8: a) Fluid temperature in a X-Z plane (y=0.3Y) with particle heat flux (W/m2). b) Fluid temperaturein a X-Z plane (y=0.3Y) with particle Nusselt number. c) Fluid velocity (|~u(x, y, z)|/U) in a X-Z plane (y=0.3Y)with particle Reynolds number for global Reynolds number = 70 and φ = 0.4 .
6.1 Local HTC
Up to now we computed HTC’s using the cup-averaged temperature computed over the full spanwise directionas given by Eq. (1). This averaging includes parts of space that are far away from the particle at hand. We alsosaw that the particle HTC’s, hp, are variable. Part of this variability is due to the fact that we do not considera local thermal driving force. In this section it will be demonstrated that the use of a locally defined thermaldriving force indeed reduces the variability of computed particle HTC’s.
Consider the particle A inside the non-homogeneous porous system in Fig. 9. In this picture a local controlvolume is drawn. This is the control volume used to compute the local gas temperature. Our simulations showthat two factors have a strong impact on the distribution of a local Nup,loc. First, the size of the control volumearound the particle and, second, the averaging method (i.e. arithmetic mean or cup-mixing temperature).According to these facts, different potential definitions of thermal driving force were analyzed.
We have considered several sizes of control volumes and finally settled on a cubic one centered at the particlewith sides of length 3 dp. For the local gas temperature different definitions can be imagined, namely: simplyaveraging over the volume, using distance dependent weight functions, or using some type of cup-averaging.We found that this last choice gave the best results. Here a local temperature is defined as the cup-averagedtemperature in the cube centered around the particle (indicated as Vf (~xp))
Tf,loc(~xp) =
∫∫∫Vf (~xp)
U(~x)T (~x)dVf∫∫∫Vf (~xp)
U(~x)dVf. (19)
Using this local temperature, the thermal driving force is the temperature difference ∆Tloc = Ts−Tf,loc andthe local particle HTC can be computed as,
hp,loc =Qp
Ap ∆Tloc. (20)
12
(b)
275
290
305
320
(a)
275
290
305
320
(b)
0
2
4
6
(c)
Figure 8: a) Fluid temperature in a X-Z plane (y=0.3Y) with particle heat flux (W/m2). b) Fluid temperaturein a X-Z plane (y=0.3Y) with particle Nusselt number. c) Fluid velocity (|~u(x, y, z)|/U) in a X-Z plane (y=0.3Y)with particle Reynolds number for global Reynolds number = 70 and φ = 0.4 .
6.1 Local HTC
Up to now we computed HTC’s using the cup-averaged temperature computed over the full spanwise directionas given by Eq. (1). This averaging includes parts of space that are far away from the particle at hand. We alsosaw that the particle HTC’s, hp, are variable. Part of this variability is due to the fact that we do not considera local thermal driving force. In this section it will be demonstrated that the use of a locally defined thermaldriving force indeed reduces the variability of computed particle HTC’s.
Consider the particle A inside the non-homogeneous porous system in Fig. 9. In this picture a local controlvolume is drawn. This is the control volume used to compute the local gas temperature. Our simulations showthat two factors have a strong impact on the distribution of a local Nup,loc. First, the size of the control volumearound the particle and, second, the averaging method (i.e. arithmetic mean or cup-mixing temperature).According to these facts, different potential definitions of thermal driving force were analyzed.
We have considered several sizes of control volumes and finally settled on a cubic one centered at the particlewith sides of length 3 dp. For the local gas temperature different definitions can be imagined, namely: simplyaveraging over the volume, using distance dependent weight functions, or using some type of cup-averaging.We found that this last choice gave the best results. Here a local temperature is defined as the cup-averagedtemperature in the cube centered around the particle (indicated as Vf (~xp))
Tf,loc(~xp) =
∫∫∫Vf (~xp)
U(~x)T (~x)dVf∫∫∫Vf (~xp)
U(~x)dVf. (19)
Using this local temperature, the thermal driving force is the temperature difference ∆Tloc = Ts−Tf,loc andthe local particle HTC can be computed as,
hp,loc =Qp
Ap ∆Tloc. (20)
12
(c)
Figure 4.6: a) Fluid temperature in a X-Z plane (y=0.3Y) with particle heat trans-fer rate (W ). b) Fluid temperature in a X-Z plane (y=0.3Y) with particle Nusseltnumber. c) Fluid velocity (|u(x, y, z)|/Us) in a X-Z plane (y=0.3Y) with particleReynolds number for global Reynolds number = 70 and φ = 0.4 .
50 Chapter 4. DNS of random arrays of monodisperse spheres
4.4.1 Local HTC
Up to now we computed HTCs using the cup-averaged fluid temperature computed
over the full spanwise direction as given by Eq. (2.1). This averaging includes parts
of space that are far away from the particle at hand. We also saw that the particle
HTCs, hp, are variable. Part of this variability is due to the fact that we do not
consider a local thermal driving force. In this section it will be demonstrated that
the use of a locally defined thermal driving force indeed reduces the variability of
computed particle HTCs.
Consider the particle A inside the non-homogeneous porous system in Fig. 4.7.
In this picture a local control volume is drawn. This is the control volume used to
compute the local gas temperature. Our simulations show that two factors have a
strong impact on the distribution of the local Nusselt number Nup,loc. First, the
size of the control volume around the particle and, second, the averaging method
(i.e. arithmetic mean or cup-mixing temperature). According to these facts, different
potential definitions of thermal driving force were analyzed.
Particle A
A control volume around particle A
Flow direction A
Figure 9: A schematic of non-homogeneous system. The test particle A and a control volume aruound it areshown.
Related to this we obtain the local particle Nusselt number as Nup,loc = hp,loc dp/k. When deciding betweenthe different possible control volumes and temperature averages we compared the standard deviation, Eq. (18),corresponding to Nup,loc. The reasoning for this criterion is that a quantity that does not vary is fully predictable,so minimizing the variability gives a quantity that is better predictable. For example, Fig. 10 compares theparticle HTC computed by using the global cup-averaged temperature with the locally defined one (in this caseRe= 30 and solids volume fraction is 0.4). Table 3 reports the relative standard deviation of Nup of the bedusing the spanwised cup-averaged temperature, Eq. (1), and the local one, Eq. (19). For all cases the localdefinition gives rise to a smaller relative standard deviation. It decreases up to a factor 2.
0 0.2 0.4 0.6 0.8 1
2
4
6
8
10
12
14
16
xX
Par
ticl
eN
uss
elt
nu
mb
er
Eq. 1Eq. 19
The average Nu of the system
Figure 10: The particle Nusselt number along the bed obtained by different type of thermal driving forces -(Re = 30 and φ = 0.4). The average Nusselt for the system is 7.63.
6.2 Local HTC correlations
By computing the particle HTC based on the local cup-averaged temperature its variability decreased signifi-cantly, but did not vanish. Not all of the influences of the local structure might be fully incorporated in thelocal temperature Tf,loc. Locally the heat flux is determined by a temperature difference and a relevant lengthscale (e.g., a boundary layer thickness). This length scale will be determined by the details of the local velocityfield and might well be correlated to a local Reynolds number and a local solids volume fraction. If part ofthe variability can, in fact, be correlated to a local Reynolds number and/or a local solids volume fractionthen the standard deviations of Nup,loc−Nup,corr(Reloc, φloc) would be significantly smaller than that of Nup,loc
13
;
Figure 4.7: A schematic of non-homogeneous system. The test particle A and acontrol volume aruound it are shown.
We have considered several sizes of control volumes and finally settled on a cubic
one centered at the particle with sides of length 3Dp. For the local gas temperature
different definitions can be imagined, namely: simply averaging over the volume,
using distance dependent weight functions, or using some type of cup-averaging. We
found that this last choice gave the best results. Here a local temperature is defined
as the cup-averaged temperature in the cube centered around the particle (indicated
4.4. Influence of micro-structure on particle-fluid heat transfer rate 51
as Vf (xp))
Tf,loc(xp) =
∫∫∫Vf (xp)
U(x)T (x)dVf∫∫∫Vf (xp)
U(x)dVf. (4.3)
Using this local temperature, the thermal driving force is the temperature differ-
ence ∆Tloc = Ts − Tf,loc and the local particle HTC can be computed as,
hp,loc =Qp
Ap ∆Tloc. (4.4)
Related to this we obtain the local particle Nusselt number as Nup,loc = hp,locDp/k.
When deciding between the different possible control volumes and temperature aver-
ages we compared the standard deviation, Eq. (5.7), corresponding to Nup,loc. The
reasoning for this criterion is that a quantity that does not vary is fully predictable,
so minimizing the variability gives a quantity that is better predictable.
For example, Fig. 4.8 compares the particle HTC computed by using the global
cup-averaged temperature with the locally defined one (in this case Re= 30 and solid
volume fraction is 0.4). Table 4.3 reports the relative standard deviation of Nup of
the bed using the spanwised cup-averaged temperature, Eq. (2.1), and the local one,
Eq. (4.3). For all cases the local definition gives rise to a smaller relative standard
deviation. It decreases up to a factor 1.3.
Table 4.3: Relative standard deviation of Nup when globally averaged and localthermal driving forces are used.
Re φ Eq. (2.1) Eq. (4.3)30 0.2 0.39 0.3450 0.2 0.33 0.30100 0.2 0.26 0.2430 0.4 0.45 0.3950 0.4 0.37 0.34100 0.4 0.31 0.2830 0.6 0.56 0.4550 0.6 0.50 0.39100 0.6 0.38 0.29
4.4.2 Local HTC correlations
By computing the particle HTC based on the local cup-averaged temperature its
variability decreased, but did not vanish. Not all of the influences of the local structure
might be fully incorporated in the local temperature Tf,loc. Locally the heat flux is
52 Chapter 4. DNS of random arrays of monodisperse spheres
Particle A
A control volume around particle A
Flow direction A
Figure 9: A schematic of non-homogeneous system. The test particle A and a control volume aruound it areshown.
Related to this we obtain the local particle Nusselt number as Nup,loc = hp,loc dp/k. When deciding betweenthe different possible control volumes and temperature averages we compared the standard deviation, Eq. (18),corresponding to Nup,loc. The reasoning for this criterion is that a quantity that does not vary is fully predictable,so minimizing the variability gives a quantity that is better predictable. For example, Fig. 10 compares theparticle HTC computed by using the global cup-averaged temperature with the locally defined one (in this caseRe= 30 and solids volume fraction is 0.4). Table 3 reports the relative standard deviation of Nup of the bedusing the spanwised cup-averaged temperature, Eq. (1), and the local one, Eq. (19). For all cases the localdefinition gives rise to a smaller relative standard deviation. It decreases up to a factor 2.
0 0.2 0.4 0.6 0.8 1
2
4
6
8
10
12
14
16
xX
Part
icle
Nuss
elt
nu
mb
er
Eq. (2.2)
Eq. (4.3)The average Nu of the system
Figure 10: The particle Nusselt number along the bed obtained by different type of thermal driving forces -(Re = 30 and φ = 0.4). The average Nusselt for the system is 7.63.
6.2 Local HTC correlations
By computing the particle HTC based on the local cup-averaged temperature its variability decreased signifi-cantly, but did not vanish. Not all of the influences of the local structure might be fully incorporated in thelocal temperature Tf,loc. Locally the heat flux is determined by a temperature difference and a relevant lengthscale (e.g., a boundary layer thickness). This length scale will be determined by the details of the local velocityfield and might well be correlated to a local Reynolds number and a local solids volume fraction. If part ofthe variability can, in fact, be correlated to a local Reynolds number and/or a local solids volume fractionthen the standard deviations of Nup,loc−Nup,corr(Reloc, φloc) would be significantly smaller than that of Nup,loc
13
Figure 4.8: The particle Nusselt number along the bed obtained by different type ofthermal driving forces - (Re = 30 and φ = 0.4). The average Nusselt for the systemis 7.63.
determined by a temperature difference and a relevant length scale (e.g., a boundary
layer thickness). This length scale will be determined by the details of the local
velocity field and might well be correlated to a local Reynolds number and a local
solids volume fraction. If part of the variability can, in fact, be correlated to a local
Reynolds number and/or a local solids volume fraction then the standard deviations
of Nup,loc − Nup,corr(Reloc, φloc) would be significantly smaller than that of Nup,loc
alone. In this formula Nup,loc is the ‘measured’ local particle HTC based on the local
thermal driving force. The second term, Nup,corr(Reloc, φloc), is a closure relation
that depends on a local Reynolds number and solids volume fraction. The optimal
definition of the local Reynolds number and solids volume fraction is the one that
minimizes the standard deviation.
In section 4.4 we defined a local Reynolds number and the comparison of Figs. 4.6b
and 4.6c suggests that the HTC is correlated to the local Reynolds number. From
profiles like that given in Fig. 6.3 we see significant deviation from the bulk for the
Nusselt number at the entrance and exit regions. Note that in these regions the solids
fraction suddenly changes from zero to the bulk value and vise-versa. The particle
heat flux in these regions might be captured to some extend by making the local
Nusselt number dependent on a local solids volume fraction. The region where the
heat flow to the particles deviates from the bulk behavior is typically larger at the
entrance than at the exit. Our DNS results show for a wide range of solids volume
4.4. Influence of micro-structure on particle-fluid heat transfer rate 53
fractions (between 0.1 and 0.6) that the HTC approaches a constant value after 2Dp
far from the entrance of subregion and after 1Dp in the exit region. This suggests that
a local solids volume fraction computed over a region that extends 2Dp upstream and
1Dp downstream might give good results. When then estimating the local particle
Nusselt number of particle i: Nuip,loc by means of a correlation that depends on local
properties Nucorr(Reiloc, φiloc) part of the fluctuations in Nuip,loc might be captured by
Nucorr(Reiloc, φiloc) through its dependence on Reiloc and φiloc.
We do not attempt to provide such a correlation here because it is of limited use.
In a coarser simulation such as DPM the specific local properties as defined above,
i.e., Reiloc and φiloc are not available. Usually the gas properties are defined on a
coarse grid and interpolated to the particle positions. Using these interpolated fields
one can define local Reiloc and φiloc at particle positions, but these are different from
those above. A similar observation holds for the temperature field and thus for the
local temperature that can be used to define Nup,loc.
Also note that it is not to be expected that the functional dependence of Nup,corr
on Reynolds number and solids volume fraction is the same as that of a correla-
tion for Nup (that is defined using spanwise integrated cup-averaged velocity in-
stead of a local one.) It is even not to be expected that 〈Nup,corr(Reiloc, φiloc)〉 =
〈Nup,corr(〈Reloc〉, 〈φloc〉) because these correlations are usually non-linear functions.
Therefore correlations obtained from macroscopic information can not be easily trans-
ferred to local correlations. All these observations imply that the best correlation for
a specific coarse simulation will depend on the details of the coarse simulation like
mesh spacings and the interpolation scheme used. Therefore the best method might
be to create tailor made correlations using DNS and with coarse graining to variables
that are defined the same as in e.g. a DPM simulation.
Besides this, even though the use of locally defined properties give HTCs that to a
certain extents predict Nuip,loc much of the detail of the DNS level is lost at the coarse
level. Therefore not all variability of Nuip,loc will be captured by a correlation. It is to
be expected that there will always be a difference between a coarse prediction Nup,corr
and the value Nuip computed in a DNS simulation. For a well chosen correlation the
mean of this deviation will be close to zero, but the standard deviation will not. The
most often used approach is to discard this variability. It might be better, however,
to model the non-resolved fluctuations using stochastic variables.
Think of the analogy with Brownian motion of colloidal particles. Forces are
exerted on such a particle due to collisions with much smaller solvent molecules. The
mean motion can be computed by equating a drag force (e.g. Stokes drag) with other
external forces. However, the fluctuating part of the motion, i.e. Brownian motion
is not captured in this case. Using stochastic variables this type of motion can be
modeled. A colloidal particle in a coarse grained Brownian dynamics simulation with
stochastic Brownian forces will follow a different path than the motion of a particle in
54 Chapter 4. DNS of random arrays of monodisperse spheres
a molecular dynamics simulation where the solvent is modelled explicitly. However,
the path in the Brownian dynamics is similar to the real one. The real motion
is not recovered, but the motion modeled by the coarse simulation is typical for a
Brownian particle. Although the Brownian motion is purely stochastic, the statistics
of the motion of the colloidal particle much more realistic compared to the case of no
stochastic force. For example when looking at a collection of colloidal particles they
exhibit the correct diffusive behavior which would not be the case otherwise.
Likewise, it might be valuable not to discard the remaining fluctuations that are
left after coarse graining the particle HTC correlation. It might be more realistic,
especially in dynamic cases such as fluidized beds, to model the variability by means
of stochastic variables. In the case of Brownian motion statistical mechanics provides
tools to obtain these stochastic terms. This is more complicated in our case. However,
the data obtained by DNS provides the statistics of the varying particle HTCs and
this might be used to also provide closure relations where the remaining fluctuations
are modeled in a stochastic manner.
4.5 Conclusion
In this study, we employed the DNS to study the heat transfer in statistically ho-
mogeneous fixed beds of particles. The physical model is constructed by a random
non-overlapping distribution of particles in a box. In order to investigate the entrance
effect on HTC, the inlet and out boundary conditions are imposed in streamwise di-
rection. The wall effect is removed by using periodic boundary condition in spanwise
direction. The non-isothermal flow through this complex geometry is solved by IBM.
With treatment of detailed information on the temperature distributions, the average
HTC of the bed is extracted as function of operating conditions. The comparison be-
tween our numerical HTC results and well-known correlation proves that the Gunn
correlation predicts the HTC of dilute system (φ < 0.3) with acceptable accuracy.
While it overpredicts significantly the HTC for dense systems (e.g. packed bed). Our
results for packed bed (φ ≈ 0.6) are in excellent agreement with prediction of Wakao
correlation. We refitted the Gunn equations, according to our results, in order to
obtain a general equation for the whole range of solids volume fraction.
Furthermore, in this chapter we have analyzed the fluctuations of the particles
HTC with respect to the average HTC of statistically homogeneous fixed beds of
particles. DNS results reveal that the particles HTC can differ significantly (up to
60%) from the average value of the bed. Since the average HTC is used for all particles
in a computational grid in the coarse scale method e.g. DPM (instead of the ‘true’
particle HTC), this deviation can affect considerably the results of simulation.
The detail thermal and flow fields show that the particle-fluid heat exchange de-
pends strongly on the heterogeneity of micro-structure near to the particle. Although
it is a difficult task to characterize the particle HTC as function of micro-structural
4.5. Conclusion 55
details, the variation of particle HTC around the average HTC of the bed can be
decreased by defining proper local Reynolds number and a thermal driving force tem-
perature. According to our results, the thermal driving force is best defined as the
cup-averaged temperature in a cube with an edge length of 3Dp around the particle.
5
CH
AP
TE
R
DNS of bidisperse spheres
Abstract
Extensive DNSs were performed to obtain the HTCs of bidisperse random arrays of
spheres. We have calculated the HTC of the bed for a range of compositions and solids
volume fractions for mixtures of spheres with a size ratio of 1:2. It was found that
the correlation of the monodisperse HTC can estimate the average HTC of bidisperse
systems well if the Reynolds and Nusselt numbers are based on the Sauter mean di-
ameter. We report the difference between HTC for each particle type and the average
HTC of the bed in the bidisperse system as function of solids volume fraction, diam-
eter ratio of particles type and Sauter mean diameter of the mixture and investigate
the heterogeneity if the individual particle HTCs.
57
58 Chapter 5. DNS of bidisperse spheres
5.1 Introduction
Many pressure drop and heat transfer correlations have been obtained for random
arrays of monodisperse spheres. In reality, the particles can have a significant size
distribution in the bed (polydispersed systems). For example, the particles can grow
as consequence of physical (coating) and chemical (polymerization) processes in flu-
idized suspensions. Therefore, these momentum and heat transfer correlations may
result in a significant error if the average deviation of the particle size from the mean
value is not negligible.
An accepted approach, in the case of non-uniform spheres, is to use an average
particle size. For example, Balakrishnan and Pei (1979) and de Souza-Santos (2004)
suggest a simple arithmetic average value and the area-volume average for the par-
ticle diameter of the bed, respectively. Despite such modifications, the numerical
simulations show significant deviation with experimental results (Shah et al. (2011)
and Benyahia (2009)). Therefore, no concrete approaches have been proposed to
characterize the heat transfer in polydisperse systems.
To date, most available correlations are based on experimental data. It is a dif-
ficult task to measure the effect of polydispersity on the drag force or heat transfer
experimentally, in particular when dense systems are of interests.
Accurate prediction of fluid-solid flows needs improved thermal and hydrodynamic
models, which require better understanding of the effect of polydispersity. Unlike mul-
tiphase experimental techniques, DNS can provide detailed information at microscale
in multiphase systems. One of the biggest advantages of DNS is that the operating
conditions can be perfectly controlled, which is often not the case in experiments.
Based on accurate numerical data from lattice-Boltzmann simulations Koch et al.
(1997) and van der Hoef et al. (2005) proposed new drag force relations. van der
Hoef et al. (2005) and Beetstra et al. (2007) conducted extensive DNS to characterize
the drag force in mono- and bidisperse arrays of spheres. They proposed a correla-
tion for the drag force applicable to both mono- and polydisperse systems, based on
the Carman-Kozeny equations. Recently, the mono- and polydisperse systems were
studied by other researchers as well (such as Yin and Sundaresan (2009) and Tenneti
et al. (2011)) and new correlations were proposed.
To the authors knowledge, no numerical investigation has been performed to in-
vestigate heat transfer in polydisperse systems of spherical particles. In this chapter,
a first step is made towards assessment of the effect of polydispersity on the heat
transfer in packed and fluidized beds. The focus of our study is on binary systems,
although it can be extended readily to polydispersed systems.
The DNS approach has been employed to simulate the bidisperse systems. The
numerical approach that we employ in this chapter is very similar to the approach
used in chapter 4 to determine the HTC in monodisperse systems. The physical model
is constructed by distribution of 54 non-overlapping bidisperse spherical particles in a
5.2. Bidisperse systems 59
cubic domain using a standard Monte Carlo procedure for hard spheres. Consequently
the DNS approach can improve our insight regarding the effect of non-ideality on the
hydrodynamic and thermal behavior of a multiphase system.
This chapter is organized as follows. First, the overall HTCs for fixed random
arrays of bidisperse systems are determined. It is shown that the results can be
described well by the refitted Gunn correlation, Eq. (4.1), if the Sauter mean diameter
is used as the effective diameter. The HTC of each particle type in a binary system
is characterized as well. The conclusion is given in the last section.
5.2 Bidisperse systems
We constructed bidisperse systems, which contain a total number N = 54 of spheres,
of which Ni spheres are of species i with diameter Di. A typical diameter ratio that
was considered is DS : DL = 1 : 2. By changing the ratio of large to small particle,
i.e. NS : NL, the influence of the composition on the heat transfer is investigated.
The geometric configurations were constructed by a random distribution of bidis-
perse spheres in a box with the aid of a Monte Carlo procedure. Fig. 5.1 shows a
typical configuration of the systems that were studied. The system shown here con-
sists of 30 small and 24 large spheres with a diameter ratio of 1:2. The solid volume
fraction is 0.5.
5.3 Results and discussion
5.3.1 Overall HTC for bidisperse systems
To date, no heat transfer correlation is available for specifically polydisperse systems.
An accepted approach is to find the corresponding monodisperse system that has
the same HTC of the polydisperse system. In other words, the Reynolds number
and solids volume fraction for the polydisperse system must be defined. Then a
general heat transfer correlation can be used for mono and polydisperse system. It
is common to consider the same solids volume fraction of polydisperse system for the
corresponding monodisperse system.
The overal Reynolds and Nusselt numbers are defined as follows:
Re = UsDe/ν, Nu = hDe/k, (5.1)
where De is the effective diameter of the polydisperse system. In fact De represents
the sphere diameter of the corresponding monodisperse system. A variety of defini-
tions for De can be found in literature. For example, Balakrishnan and Pei (1979)
and de Souza-Santos (2004) suggest a simple arithmetic average value and an average
based on surface area and particle volume for polydisperse system, respectively.
The HTC depends on the available surface area of the bed. Therefore, according to
this criterion, the corresponding monodisperse system possesses the same volume over
60 Chapter 5. DNS of bidisperse spheres
Figure 5.1: Example of a bidisperse system used in DNS. N=54, NL = 24, DL/DS =2, φ=0.5.
surface area ratio as the polydisperse system. In this study we tested the suitability
of the Sauter mean diameter DSau to estimate the proper De of the polydisperse
system.
DSau =
∑iNiD
3i∑
iNiD2i
, (5.2)
where Ni is the number of particles with diameter Di. In order to test this approach,
the Nusselt numbers were computed for fixed arrays of bidispersed spherical particles.
To obtain the overall Nusselt number the same procedure is used as before. First
the bed is divided into slices (perpendicular to the flow direction). The slices are
taken so thin that the cup-averaged temperature does not change significantly. The
thermodynamic driving force is taken to be the particle temperature minus the cup-
averaged temperature corresponding to the slice, let’s call this ∆Tslice. Next, the total
heat flow through the particle surfaces in that slice is to the fluid is computed, let’s
call this Qslice. To compute the heat transfer coefficient we compute
hslice =Qslice
ap Vslice∆Tslice. (5.3)
5.3. Results and discussion 61
Here ap is the specific surface area (which equals 6φ/DSau with φ the solid volume
fraction). The overall HTC is then computed as the average of hf,slice over the slices
(excluding those at the entry and exit).
Figs. 5.2 and 5.3 show that Nusselt numbers of bidisperse system obtained by
DNS together with results obtained from the modified Gunn correlation (Eq. 4.1).
The Nusselt numbers are obtained for solid volume fractions φ = 0.4, 0.5 and 0.6
and a range of Reynolds numbers and volume fraction of large particles. The results
are compared with results for fixed arrays of monodispersed spherical particles. It
is observed that the DNS results of mono- and bidisperse systems agree well with
each other for the investigated operations conditions. Therefore, our results indicate
that the overal Nusselt number for a bed with with mono and bidispersed spherical
particles can be correlated with a general equation if provided that the effective
diameter of the bidisperse system is based on the Sauter mean diameter.
5.3.2 The species HTC in bidisperse systems
In simulations of polydisperses system using coarse-grained approaches, the HTC of
each type of particle is required. The local HTC is computed from the heat flow rate
from the particle, the temperature difference between the particle, the cup-averaged
temperature at the stream-wise position of the particle’s center, and its area,
hi =Qi
πD2i∆Ti
and Nui =hiDi
k. (5.4)
Note that we choose to use the particle diameter to define the particle Nusselt number.
Contrary to the experimental methods, the true value of Nui can be readily obtained
with the aid of DNS.
In our case we consider a bidisperse mixture of large and small species. The
average Nusselt number per species is,
NuL =1
NL
NL∑
j=1
NuL,j and NuS =1
NS
NS∑
j=1
NuS,j . (5.5)
Note that the overall Nusselt number can be obtained from the total heat flow rate
by summing the heat flow rates of the small and large particles. In this case the
particle diameter in definition Eq. (5.4) needs to be taken into account to go from
species Nusselt number to species heat transfer coefficient. Next, the area needs to
be taken into account. The effective diameter enters form the definition Eq. (5.1).
The formula for the overall HTC becomes,
Nuav =
∑iNi(Ai/Di)Nui∑iNi(Ai/De)
=
∑iNiDiNui∑iNi(D
2i /De)
(5.6)
Here Ni is the number of particles of species i (i = L, S in our case). Note that the
derived formula is general. For our choice we need to substitute De = DSau.
62 Chapter 5. DNS of bidisperse spheres
(a)
(b)
Figure 5.2: The Nusselt number of a bidispersed systems in comparison to themonodisperse system and prediction on the basis of the modified Gunn correlation.The diameter ratio and total number of particles are 2 and 54, respectively. a) NL=6b) NL=14.
5.3. Results and discussion 63
(a)
(b)
Figure 5.3: The Nusselt number of a bidispersed systems in comparison to themonodisperse system and prediction on the basis of the modified Gunn correlation.The diameter ratio and total number of particles are 2 and 54, respectively. a) NL=18b) NL=24.
64 Chapter 5. DNS of bidisperse spheres
Figure 5.4: A parity plot where the two overall Nusselt numbers are compared fordifferent Reynolds number (φ=0.5).
Note that there exist two ways to compute the overall Nusselt number, namely,
by means of the overall HTC defined by Eq. (5.3) and via the averaging of individual
particle Nusselt numbers, Eq. (5.5). For large systems these two ways are expected
to give the same value. Since, however, small systems are considered this is not
necessarily the same here. Fig. 5.4 shows a parity plot where the two overall Nusselt
numbers are compared. The deviation from x = y indicates the level of error due to
the finite size of the systems considered.
With the aid of the DNS approach, we are able to obtain individual particle Nusselt
numbers, Nuj , at different operating conditions. These can next be used to compute
the species Nusselt numbers and the overall overaged Nusselt number. Figs. 5.5, 5.6
and 5.7 presents the Nuav, NuS and NuL (subscripts L and S refer to the large and
the small spheres, respectively) for a range of Reynolds numbers and compositions at
several solids volume fractions. These figures show that (as expected) all the three
Nusselt numbers, Nuav, NuL and NuS , increase with the Reynolds number.
5.3.3 Heterogeneity of heat transfer in bidisperse systems
Thus far, the discussion was concerned with the average Nusselt numbers of species
i in bidisperse systems. In Fig. 5.8 two histograms are shown for the occurrence
of individual Nusselt numbers in two typical systems. The individual particles are
5.3. Results and discussion 65
Figure 5.5: Particle Nusselt number Nu, NuL, NuS when φ=0.4.
Figure 5.6: Particle Nusselt number Nu, NuL, NuS when φ=0.5.
66 Chapter 5. DNS of bidisperse spheres
Figure 5.7: Particle Nusselt number Nu, NuL, NuS when φ=0.6.
categorized in species. It is clear from this figures that the histograms are broad, but
not fully overlapping for two species.
The variation of the particle Nusselt number NuS,j around the mean Nusselt
number of each species in bidisperse systems can be quantified by the relative standard
deviation as,
σrel,S =1
NuS
√∑NS
j=1(NuS,j −NuS)2
NS − 1, (5.7)
σrel,L =1
NuL
√∑NL
j=1(NuL,j −NuL)2
NL − 1, (5.8)
Figs. 5.9, 5.10 and 5.11 shows the values of σrel,L and σrel,S as a function of φ
and the Reynolds number. From this figure it can be seen that σrel,S and σrel,L
increase with decreasing Re and increasing φ. σrel,S and σrel,L are as high as 60%
and 86%, respectively, in the most extreme cases. This deviation originates from
the heterogeneity in the microstructure in vicinity of the particle. The effect of
heterogeneity of microstructure is discussed in more detail in chapter 4.
The deviation for drag and HTC was characterized in Kriebitzsch et al. (2013)
and this thesis (chapter 4), respectively. They tried to relate this deviation for each
particle to the local microstructure information. For example, Kriebitzsch estimated
5.3. Results and discussion 67
0 5 10 15 20 250
0.1
0.2
0.3
Nup
Probab
ilitydensity
function
Nuip,SNuip,L
Figure 10: Distribution of the particle Nusselt number in the bed (Re=100, φ=0.4, NL = 14).
0 5 10 15 200
0.1
0.2
0.3
Nup
Probab
ilitydensity
function
Nuip,SNuip,L
Figure 11: Distribution of the particle Nusselt number in the bed (Re=70, φ=0.4, NL = 14).
5
(a)
0 5 10 15 20 250
0.1
0.2
0.3
Nup
Probabilitydensity
function
Nuip,SNuip,L
Figure 10: Distribution of the particle Nusselt number in the bed (Re=100, φ=0.4, NL = 14).
0 5 10 15 200
0.1
0.2
0.3
Nup
Probab
ilitydensity
function
Nuip,SNuip,L
Figure 11: Distribution of the particle Nusselt number in the bed (Re=70, φ=0.4, NL = 14).
5
(b)
Figure 5.8: Distribution of the particle Nusselt number in the bed. a) Re=100, φ=0.4,NL = 14. b) Re=70, φ=0.4, NL = 14.
68 Chapter 5. DNS of bidisperse spheres
Figure 5.9: σrel,L, σrel,S when φ=0.4.
that the particle drag coefficient based on the local solids volume fraction for each
particle φl, using Voronoi tessellation method. However, Kriebitzsch stated that the
deviation increases if the local solids volume fraction is used in the estimate of the
particle drag coefficient.
In chapter 4 the fluctuation of particle HTC with respect to the average value of
the bed was characterized using DNS for random arrays of equal sized spheres. It was
concluded that the fluctuation might be modeled in a stochastic manner. Although
no stochastic model was proposed, it was shown that the variation of particle HTC
around the average HTC of the bed can be decreased by defining a proper local
Reynolds number and thermal driving force.
We observe that the local microstructure effect in bidisperse systems is more pro-
nounced in comparison to monodisperse systems. More extensive studies are required
to quantify the relation between the local microstructural information and local par-
ticle HTC in a bidisperse system. This is, however, beyond the scope of this study.
5.4 Conclusions
In this study, DNS is employed to study the heat transfer in stationary arrays of
bidisperse spheres. The physical model is constructed by a random distribution of
5.4. Conclusions 69
Figure 5.10: σrel,L, σrel,S when φ=0.5.
Figure 5.11: σrel,L, σrel,S when φ=0.6.
70 Chapter 5. DNS of bidisperse spheres
bidisperse spheres in a box. On the basis of detailed analysis of the computed temper-
ature distributions, the average HTC of the bed is determined as function of operating
conditions. Our results reveal that the average HTC of binary systems can be de-
termined according to the heat transfer correlation for monodisperse system if the
Reynolds and Nusselt number are based on the Sauter mean diameter. Based on our
DNS results, we characterized the HTC of each species of particles in a binary system
as function of solids volume fraction and diameter ratio.
In addition, the fluctuations of the particles HTC of each type with respect to
the average HTC of type i in a bidisperse system was quantified. Our DNS results
reveal that the particles HTC can differ up to 60% from the average value of the
particles. This altogether indicates that the particle HTC depends strongly on the
microstructural heterogeneity of microstructure in the vicinity of the particle.
6
CH
AP
TE
R
DNS of non-spherical particles
Abstract
DNS are conducted to characterize the fluid-particle HTC in fixed random arrays of
non-spherical particles. The objective of this study is to examine the applicability
of well-known heat transfer correlations, that are proposed for spherical particles,
to systems with non-spherical particles. In this study sphero-cylinders are used to
pack the beds and the non-isothermal flows are simulated by employing the IBM.
The simulations are performed for different solids volume fractions and particle sizes
over low to moderate Reynolds numbers. According to the detailed heat flow pattern,
the average HTC is calculated in terms of the operating conditions. The numerical
results show that the heat-transfer correlation of spherical particles can be applied to
all test beds by choosing a proper effective diameter used in the correlations for the
non-spherical particles. Our results reveal that the diameter of sphero-cylinder is the
proper effective diameter for characterizing the heat transfer.
71
72 Chapter 6. DNS of non-spherical particles
6.1 Introduction
Most of the pressure drop and HTC correlations are obtained for randomly packed
beds of spheres. For example, it is generally accepted that the pressure drop in a
porous bed packed with spherical particles can be estimated reasonably well from the
Ergun correlation. The Ergun equation relates the pressure drop to the particle size
and the bed porosity. However, the extension of the Ergun equation to a bed packed
with non-spherical particles is not straightforward.
On the other hand, recent numerical studies (Freund et al. (2003), Guardo et al.
(2006), Nijemeisland (2000)) indicate that not only the local behavior but also the
macroscopic quantities, such as the pressure drop, are significantly affected by local
micro-structural properties of the bed.
An accepted approach which has its foundation in the Carman-Kozeny approxima-
tion, is to use an effective diameter for non-spherical particles in the Ergun equation.
Nemec and Levec (2005) have fitted the pressure drop for particles of different shapes
to the generalized Ergun equation. They concluded that the use of a general effective
diameter is not sufficient to capture the effect of non-spherical particle shapes. They
proposed to evaluate the constants of the Ergun equation as a function of the particle
size.
Our current knowledge of the heat transfer characteristic for these systems follows
mainly from experiments (e.g.Gunn (1978), Wakao et al. (1979) and Kunii and Lev-
enspiel (1991) obtained from random beds with spherical particles. Although these
well-known correlations are widely used to predict the heat transfer characteristics
of packed and fluidized beds, their applicability to structured beds or non-spherical
particles has not been assessed yet. Calis et al. (2001) and Romkes et al. (2003)
characterized the momentum and heat transfer in five different types of composite
structured packed beds of spheres using numerical and experimental methods. These
results revealed that macroscopic flow and heat transfer characteristics are affected
significantly by packing features. Yang et al. (2010) studied the effects of packing
form and particle shape on the flow and heat transfer characteristics in structured
packed beds. They found that, with proper selection of packing form and particle
shape, the hydrodynamic and thermal performance in structured packed beds can be
greatly improved. Their results show that the correlations for momentum and heat
transfers extracted from random packings overpredict the pressure drops and HTC
for structured packings.
All these results indicate that the heat transfer between fluid and particles is
strongly affected by the local flow structure and varies spatially for non-uniform
structures. Therefore, the effect of the packing material shape needs to be considered
for accurate prediction of fluid-particle heat transfer characteristics.
Pressure drop or heat transfer correlations can be estimated with the aid of numer-
ical experiment as function of operating conditions. This approach has been used by
6.2. Physical model and numerical method 73
Hill et al. (2001) and Beetstra et al. (2007) to characterize the pressure drop in beds
of spherical particles over a wide range of Reynolds number and solids volume frac-
tion. Dorai et al.(2014) performed DNS of flows through fixed beds of monodisperse
and polydisperse spherical and cylindrical particles. They investigated the influence
of the particle shape and the degree of poly-dispersity on the pressure drop through
the fixed bed in the viscous regime.
Recently, this approach has been extended to heat transfer problems (Deen et al.
(2012), Tenneti et al. (2013), Tavassoli et al. (2013)) and the Nusselt number for sta-
tionary arrays of spherical particles was estimated with the aid of DNS. These results
show significant deviations between computed Nusselt number and predictions on
basis of heat transfer correlations even in beds with monodisperse spherical particles.
These findings motivate us to examine the heat transfer in a random array of non-
spherical particles. According to authors’ knowledge, no studies have been published
in this area yet and this study provides the first results of heat transfer in such
systems. In this study, we employed the IB method to simulate non-isothermal flow
through random fixed arrays of sphero-cylinders. The physical model is constructed
by a random distribution of non-overlapping sphero-cylinders in a cubic domain by a
standard Monte Carlo procedure for hard cylinders. Based on the predicted flow and
temperature fields, the average heat transfer coefficient was computed as function of
particle shape, Reynolds number and porosity.
The chapter is organized as follows. First, the definition of the effective diameter
is detailed. Then the computed HTCs for stationary arrays sphero-cylinders of are
also compared with well-known correlations, the effect of particle size on HTC is
investigated as well. In the final section the conclusions are given.
6.2 Physical model and numerical method
6.2.1 Physical model
The simulation approach is identical as described in section 2.6.1 for stationary arrays
of spheres. The non-spherical particle used in present study is the sphero-cylinder
with diameter Dp and length Lp (Fig. 6.1). Three different aspect ratios (Lp/Dp)
of 2,3 and 4 are considered to investigate the effect of the aspect ratio on the fluid-
particle heat transfer.
As shown in Fig. 6.1, the computational domain consists of inlet, packed and
outlet sections. The packed section was created by a random distribution of N = 30
non-overlapping sphero-cylinders, with random orientation, in a 3-dimensional duct
using a standard Monte Carlo method. The sizes of inlet and outlet sections are
equal and set to Lp×Lps×Lps for all simulations (where Lps is the length of packed
section).
Some simulations are performed on different grid sizes to ensure that the results
are mesh independent (i.e. when the deviation is lower than 3%). Table 6.1 reports
74 Chapter 6. DNS of non-spherical particles
(a)
(b)
Figure 6.1: (a) A typical particle configuration used in the simulations. (b) Repre-sentation of a sphero-cylinder surface with Lagrangian points.
6.3. Heat transfer correlations in packed and fluidized beds 75
the employed mesh sizes in our simulations of arrays of sphero-cylinders.
Table 6.1: The ratios of particle diameter to the grid size in order to obtain meshindependent results. These ratios are the same for all aspect ratios
Re, φ 0.1 0.2 0.3 0.4 0.5 0.6
10 20 20 30 30 30 3030 20 20 30 30 30 3050 30 30 40 40 40 4070 30 30 40 40 40 40100 30 30 40 40 40 40
.
6.3 Heat transfer correlations in packed and
fluidized beds
The Gunn and Wakao correlations were developed on basis of published experimental
data for both spherical and cylindrical particles. However, Gunn and Wakao did not
define clearly the Reynolds number and Nusselt numbers for non-spherical particles.
Therefore, the extension of these two correlations to cylindrical particles, or non-
spherical particles in general, is not straightforward. The values of the Nusselt number
and the Reynolds number depend on the definition used for the effective diameter,
De,
Nu = hDe/k, and Re = UsDe/ν (6.1)
For a spherical particle, De is the diameter of the particle. However, the definition
of De for non-spherical particles is ambiguous. In addition, the definitions of De can
be different in calculations of Reynolds number and Nusselt numbers.
Nsofor and Adebiyi (2001), Bird et al. (2007), Yang et al. (2010) and Incropera
(2011) proposed the HTC correlations in a packed bed consisting of the non-spherical
particles. They defined the dimensionless numbers as follows:
Re = UsDh/ν, Dh =VpApφ
Nu = hDeq/k, Deq = equivalent particle diameter =
(6Vpπ
)1/3 (6.2)
Then they obtained different values for the constants of their correlations for each
type of non-spherical particle, which is not convenient.
76 Chapter 6. DNS of non-spherical particles
On the other hand it is well-known that the Gunn or Wakao equations fit the
experimental data of spherical-particle beds well. Now the intriguing question is
whether these equations can be used, using the same constant parameters, for a bed
with non-spherical particles. In this case the question arises which effective diameter
should be used in the calculation.
To answer these questions, an extensive set of DNS was performed to investigate
the fluid-particle heat transfer characteristics in random fixed arrays packed with
sphero-cylinder particles. By assuming the validity of Eq. 4.1 for non-spherical
particles, an effective particle diameter is deduced from Eq. 4.1 and the computed
HTC in the bed. The definition of the effective diameter is discussed in the following
section.
6.3.1 Effective diameters of non-spherical particles
Two often used effective diameter definitions, that can be computed from geometric
properties of a particle, are the Sauter mean diameter Ds and the equivalent volume
sphere diameter Deq,
Ds =6VpAp
and Deq =
(6Vpπ
)1/3
, (6.3)
respectively, where Vp and Ap are the volume and surface area of the particle. Both
definitions give the diameter for a spherical particle, and both definitions scale linearly
in case the size of a particle is scaled equally in all special directions. For non-spherical
particles both parameters, scale differently when changing the aspect ratio.
Since the Gunn and Wakao correlations are based on beds with spherical and
cylindrical particles, the diameter of the cylinder can be used as an effective diameter
(De = Dp) as well. Therefore, the sphero-cylinder diameter can be another option
to be used as effective diameter. In this study, these 3 definitions are considered,
De = Ds, Deq and Dp, to define dimensionless numbers.
6.4 Results and discussion
6.4.1 The local heat transfer coefficient in the bed
In Fig. 6.2 an example of the distributions of non-dimensional temperature ((T (x, y, z)−T∞)/(Ts − T∞)) and non-dimensional velocity magnitude (|u|/Us) of the fluid inside
a random array of sphero-cylinders when Re= Us ·Dp/ν = 10, Lp/Dp=2 and φ = 0.3
is shown.
The HTC along the bed can be obtained from Eq. 2.2 according to computed
velocity and temperature distributions in each cross section. Fig. 6.3 shows the
profile of local Nusselt number along the flow direction in the bed. The profile of
local Nusselt number is obtained from averaging over 3 different configurations to
obtain a better estimate of the average Nusselt number.
6.4. Results and discussion 77
0
0.25
0.5
0.75
1
(a)
0
0.75
1.5
2.25
2.56
(b)
Figure 2: Computed fluid [a] non-dimensional temperature ((T (x, y, z) − T∞)/(Ts − T∞)) and [b] non-dimensional velocity magnitude (|~u|/Us) distributions in a random array of sphero-cylinderical particles.(Re=10.0, L/D=2 and ε=0.7)
in all special directions. For non-spherical particles both parameters, scale differently when changing theaspect ratio.
Since the Gunn and Wakao correlations are based on beds with spherical and cylindrical particles, di-ameter of cylinder can be used as an effective diameter (De = Dp) as well. As far as the author know,the heat transfer correlations have not been analyzed for cylindrical particles based on employing diameterof particle as effective diameter. In this study, these 3 definitions are employed, De = Ds, Deq and Dp, todefine dimensionless numbers.
3.2 Heat transfer coefficient in packed and fluidized bed
The local forced convection HTC in a bed is defined as:
h(x) =Q(x)
Ts − Tb(16)
where Q(x) is the local heat flux (W/m2) and Tb is the cup-mixing temperature of the fluid and defined as:
Tb =
∫Acu(x, y, z)T (x, y, z) dydz∫
Acu(x, y, z) dydz
(17)
where Ac is the cross sectional area of the domain.By employing the analogy with thermally fully developed flow in pipes, it is expected that h(x) is
independent of x for a statistically homogeneous system. In this study, the average HTC of the bed incalculated as the mean value of h(x) along the bed.
4 Results and discussion
4.1 The mean heat transfer coefficient in a fixed random array of sphero-cylinders
In Fig. 2 an example of the distributions of non-dimensional temperature ((T (x, y, z)−T∞)/(Ts−T∞)) andnon-dimensional velocity magnitude (|~u|/Us) of the fluid inside a random array of sphero-cylinders whenRe=10, L/D=2 and ε = 0.7 is shown.
6
(a)
0
0.25
0.5
0.75
1
(a)
0
0.75
1.5
2.25
2.56
(b)
Figure 2: Computed fluid [a] non-dimensional temperature ((T (x, y, z) − T∞)/(Ts − T∞)) and [b] non-dimensional velocity magnitude (|~u|/Us) distributions in a random array of sphero-cylinderical particles.(Re=10.0, L/D=2 and ε=0.7)
in all special directions. For non-spherical particles both parameters, scale differently when changing theaspect ratio.
Since the Gunn and Wakao correlations are based on beds with spherical and cylindrical particles, di-ameter of cylinder can be used as an effective diameter (De = Dp) as well. As far as the author know,the heat transfer correlations have not been analyzed for cylindrical particles based on employing diameterof particle as effective diameter. In this study, these 3 definitions are employed, De = Ds, Deq and Dp, todefine dimensionless numbers.
3.2 Heat transfer coefficient in packed and fluidized bed
The local forced convection HTC in a bed is defined as:
h(x) =Q(x)
Ts − Tb(16)
where Q(x) is the local heat flux (W/m2) and Tb is the cup-mixing temperature of the fluid and defined as:
Tb =
∫Acu(x, y, z)T (x, y, z) dydz∫
Acu(x, y, z) dydz
(17)
where Ac is the cross sectional area of the domain.By employing the analogy with thermally fully developed flow in pipes, it is expected that h(x) is
independent of x for a statistically homogeneous system. In this study, the average HTC of the bed incalculated as the mean value of h(x) along the bed.
4 Results and discussion
4.1 The mean heat transfer coefficient in a fixed random array of sphero-cylinders
In Fig. 2 an example of the distributions of non-dimensional temperature ((T (x, y, z)−T∞)/(Ts−T∞)) andnon-dimensional velocity magnitude (|~u|/Us) of the fluid inside a random array of sphero-cylinders whenRe=10, L/D=2 and ε = 0.7 is shown.
6
(b)
Figure 6.2: Computed fluid [a] non-dimensional temperature ((T (x, y, z)−T∞)/(Ts−T∞)) and [b] non-dimensional velocity magnitude (|u|/Us) distributions in a randomarray of sphero-cylinderical particles. (Re= Us ·Dp/ν = 10, Lp/Dp=2 and φ = 0.3)
78 Chapter 6. DNS of non-spherical particles
Due to the statistical homogeneity of the Nusselt number in the flow direction,
the average Nusselt number of the bed is obtained by averaging Nu(x) along the flow
direction. As it is observed, the local Nusselt number fluctuates about a constant
value (i.e the average HTC of the bed). This fluctuation originates from the variation
of fluid-particle interface in the cross-sectional planes along the flow direction.
Figure 6.3: The local Nusselt number Nu(x) along the flow direction in a bed forRe= Us ·Dp/ν = 10 and φ = 0.1.
6.4.2 The influence of effective diameter on Reynolds and
Nusselt numbers
In total, 270 simulations were conducted (3 aspect ratios, 5 Reynolds numbers, 6 solids
volume fractions, 3 independent configurations) to obtain the HTC of the beds. After
the HTC of a specific simulation is obtained, the corresponding Reynolds and Nusselt
numbers must be quantified according to Eq. 6.1. As mentioned before, the Reynolds
and Nusselt numbers are function of the adopted definition for De. Therefore, for
each simulation different Reynolds and Nusselt numbers are obtained according to
the adopted definition of De. For example for a specific simulation, Nusselt numbers
are h ·Dp/k, h ·Ds/k and h ·Deq/k if Dp, Ds and Deq are used as effective diameter,
respectively. The proper effective diameter for the evaluation of the Nusselt number
6.4. Results and discussion 79
is the one that gives the Nusselt numbers with the best agreement with the empirical
correlation. The Reynolds numbers for a specific simulation are Us ·Dp/ν, Us ·Ds/ν
and Us ·Deq/ν if Dp, Ds and Deq are used as effective diameter, respectively.
Figs. 6.4, 6.5 and 6.6 show the computed Nusselt numbers using these three
definitions of De. These 3 plots correspond to the same simulations. But since
different effective diameters are used, for each simulation, different Reynolds and
Nusselt numbers are obtained.
6.4.3 The mean heat transfer coefficient in a fixed random array
of sphero-cylinders
Figs. 6.4, 6.5 and 6.6 show Nusselt numbers obtained by the DNS, for fixed random
arrays of sphero-cylinders, and the modified Gunn correlation (Eq. 4.1). Eq. 4.1
represents the Nusselt number for spherical particles. In fact, we compare the Nusselt
numbers of sphero-cylinder and spherical particles together. If they match with each
other, then a correlation can be used for both types of particles.
Figs. 6.4 and 6.5 show that the Nusselt number is overestimated when the Sauter
mean or equivalent diameter is taken as the effective particle diameter. However in
Fig. 6.6 the Nusselt numbers were well-predicted if the effective particle diameter is
represented by the diameter of the sphero-cylinder particles.
In all figures, an overall agreement is observed in the trends of the computed and
the experimental results. However, it seems that the parameters of Eq. 4.1 must be
modified for sphero-cylinders when the Sauter mean and equivalent diameters were
taken as the effective particle diameter (on contrary to the case when the cylinder
diameter of the sphero-cylinders is used as effective diameter). Therefore, one cor-
relation can be used for both spherical and sphero-cylindrical particles with proper
selection of the effective diameter.
The modified Gunn correlation gives the Gunn and Wakao predictions at low and
high solids volume fraction, respectively. Therefore, we can conclude that the Gunn
and Wakao correlations can be used for low (φ < 0.3) and high (φ ≈ 0.6) solids
volume fraction, respectively, in a bed with sphero-cylinder particles if the Reynolds
and Nusselt numbers are defined according to the cylinder diameter of the sphero-
cylinders.
6.4.4 The effect of particle shape on the mean heat transfer
coefficient
The effect of the particle shape on the Nusselt number is shown as well in Figs. 6.4,
6.5 and 6.6. The Nusselt numbers that are reported in figures by circle, square and
triangle symbols indicate aspect ratios of 2,3 and 4, respectively. Figs. 6.4 and 6.5
show that the Nusselt numbers of particles with different aspect ratios, at the same
solids volume fraction, collapse on the same master curves.
80 Chapter 6. DNS of non-spherical particles
20 40 60 80 100 120 140 160 180
5
10
15
20
25
30
35
40
Re
Nuss
elt
No.
Eq. 4.1 (φ = 0.1)
Eq. 4.1 (φ = 0.2)
Eq. 4.1 (φ = 0.3)
Eq. 4.1 (φ = 0.4)
Eq. 4.1 (φ = 0.5)
Eq. 4.1 (φ = 0.6)
Simulation (L/D=2)
Simulation (L/D=3)
Simulation (L/D=4)
Figure 2: The mean Nusselt number in random arrays of sphero-cylinder obtained from modified Eq. 10correlation and numerical simulations when De = Deq. The DNS results are shown by circles, squares andtriangle when aspect ratio is 2,3 and 4, respectively.
3
Figure 6.4: The mean Nusselt number in random arrays of sphero-cylinder obtainedfrom modified Gunn correlation (Eq. (4.1)) and numerical simulations when De =Deq. The DNS results are shown by circles, squares and triangle when the aspectratio equals 2,3 and 4, respectively.
It is observed in Fig. 6.6 that the Nusselt number of the bed is almost the same for
different aspect ratios of particles (at the same Reynolds number and solids volume
fraction) if Dp is used as effective diameter. Since the Sauter mean and equivalent
diameters are affected by aspect ratio, the Nusselt numbers are not the same for
different aspect ratio in Figs. 6.4 and 6.5.
Our results show that the aspect ratio (or in general the shape factor for a non-
spherical particle) plays an insignificant role in the laminar flow regime and De can
cover the influence of particle shape on HTC. This conclusion is contrary to the
friction factor prediction in a packed and fluidized beds, where the shape factor has a
significant effect on pressure drop. In other words, heat transfer in a bed depends on
the available surface area of the particles while pressure drop is sensitive to particle
size and volume.
6.5. Conclusion 81
20 40 60 80 100 120 140
5
10
15
20
25
30
35
Re
Nuss
elt
No.
Eq. 4.1 (φ = 0.1)
Eq. 4.1 (φ = 0.2)
Eq. 4.1 (φ = 0.3)
Eq. 4.1 (φ = 0.4)
Eq. 4.1 (φ = 0.5)
Eq. 4.1 (φ = 0.6)
Simulation (L/D=2)
Simulation (L/D=3)
Simulation (L/D=4)
Figure 3: The mean Nusselt number in random arrays of sphero-cylinder obtained from modified Eq. 10correlation and numerical simulations when De = Ds. The DNS results are shown by circles, squares andtriangle when aspect ratio is 2,3 and 4, respectively.
4
Figure 6.5: The mean Nusselt number in random arrays of sphero-cylinder obtainedfrom modified Gunn correlation (Eq. (4.1)) and numerical simulations whenDe = Ds.The DNS results are shown by circles, squares and triangle when the aspect ratioequals 2,3 and 4, respectively.
6.5 Conclusion
In this study, the DNS approach was used to investigate the heat transfer in fixed
random arrays of non-spherical particles. The random fixed array was constructed
by a distribution of sphero-cylinders in a cubic domain. The non-isothermal flows
through the arrays were computed using IBM. The computed heat transfer coefficients
of the beds were characterized in terms of the operating conditions. The numerical
results indicate that the heat-transfer correlation of spherical particles can be used
for sphero-cylinders particles if the cylinder diameter of the sphero-cylinders is used
in the definitions of Reynolds and Nusselt number. Our results show that the aspect
ratio (or in general the shape factor for a non-spherical particle) plays an insignificant
role in characterizing the HTC of the bed.
82 Chapter 6. DNS of non-spherical particles
10 20 30 40 50 60 70 80 90 1002
4
6
8
10
12
14
16
18
20
22
24
Re
Nuss
elt
No.
Eq. 4.1 (φ = 0.1)
Eq. 4.1 (φ = 0.2)
Eq. 4.1 (φ = 0.3)
Eq. 4.1 (φ = 0.4)
Eq. 4.1 (φ = 0.5)
Eq. 4.1 (φ = 0.6)
Simulation (L/D=2)
Simulation (L/D=3)
Simulation (L/D=4)
Figure 1: The mean Nusselt number in random arrays of sphero-cylinder obtained from modified Eq. 10correlation and numerical simulations when De = Dp. The DNS results are shown by circles, squares andtriangle when aspect ratio is 2,3 and 4, respectively.
2
Figure 6.6: The mean Nusselt number in random arrays of sphero-cylinder obtainedfrom modified Gunn correlation (Eq. (4.1)) and numerical simulations whenDe = Dp.The DNS results are shown by circles, squares and triangle when the aspect ratioequals 2,3 and 4, respectively.
Acknowledgements
The author wants to thank Marjolein Dijkstra and Ran Ni from the University of
Utrecht for providing us with the configurations of densely packed spherocylinders.
7
CH
AP
TE
R
Summary and recommendations
83
84 Chapter 7. Summary and recommendations
7.1 Summary and general conclusions
Heat transfer in gas-solid system are frequently encountered in many processes such
as the chemical, petrochemical, metallurgical and food processing industries. The
behaviors of such systems can be predicted with employing the CFD techniques.
However, detailed simulation of these systems is computationally expensive since
the heat transfer in multiphase flow is very complex due to a wide range of time-
and length-scales involved. With the help of multi-scale modeling approach, multi-
phase simulations can be done in a reasonable time but with less detail and accuracy.
However, the multi-scale approach requires closure equations for the modeling of un-
resolved sub-grid phenomena.
The energy transfer between gas and solid phases is represented by the HTC. The
HTC can be obtained from analytical theory, experiments and DNS, each with their
own strong and weak points. Typically the macroscopic transport properties, like
HTC, are affected significantly by the micro-structural details like solid volume frac-
tion, particle size, particle shape and physics of the transport processes. Therefore, it
is necessary to study the flow characteristics at microscale level to gain more insight
at the macroscopic transport properties.
This thesis focuses on the derivation of heat transfer correlations for random arrays
of spherical and non-spherical from fully resolved simulations. First, we extended the
IBM proposed by Uhlmann to heat transfer problems and demonstrated that it is
a viable method for understanding transport phenomena in gas-solid flows. Then
according to developed thermal model and obtained detail thermal fields, the HTC
was measured numerically.
In particular, we investigated the effect of several micro-structural parameters,
such as particle shape and distribution, on the mean HTC of the bed over a range
of small to moderate Reynolds number. The results are given in the form of a gen-
eral heat transfer correlation valid at all solid volume fractions, which can readily
be employed by the coarse-grained methods (e.g. Two-Fluid and Discrete Element
Methods). We expect that these new results will improve the modeling of gas-solid
systems using coarse-grained methods.
The non-isothermal flows in both random mono and bidisperse gas-solid systems
were studied using DNS. We compared the numerical HTC of monodisperse systems
with well-known heat transfer correlations proposed by Gunn and Wakao. Our results
prove that the Gunn and Wakao equations predict the HTC of dilute (φ < 0,3) and
dense systems (φ < 0.6) well, respectively. We refitted the Gunn equation, according
to our results, in order to obtain a general equation for the whole range of solid
volume fraction. Furthermore, we analyzed the fluctuations of the particles HTC
with respect to the average HTC of the fixed beds of monodisperse particles. These
results show that the particles HTC can differ up to 60% from the mean HTC of the
bed.
7.2. Outlook and recommendations 85
This analysis of fluctuations reveals the strong effect of the heterogeneity of the
micro-structure near the particle on the particle HTC. By defining properly the local
Reynolds number and thermal driving force temperature, the variation of particle
HTC around the mean HTC of the bed can be decreases. According to simulation
results, we suggest the proper thermal driving force as the cup-averaged temperature
in a cube with an edge length of 3Dp around the particle.
This thermal model is applied to bidisperse gas-solid system as well. We found out
that heat transfer correlation of monodisperse system is valid for bidisperse system
if the Reynolds and Nusselt numbers are based on the Sauter mean diameter. In
addition, the detailed results show that the local microstructure effect in bidisperse
systems is more pronounced in comparison to monodisperse systems.
DNS were performed to characterize the HTC in fixed random arrays of non-
spherical particles. In this thesis sphero-cylinders were used to construct the bed.
The simulations were performed for a wide range of solids volume fractions and par-
ticle sizes over low to moderate Reynolds numbers. The results of simulations show
that the heat-transfer correlation of spherical particles can be applied to a bed with
sphero-cylinders if the diameter of sphero-cylinder is used as effective diameter in the
correlation.
7.2 Outlook and recommendations
Although we discussed several aspects of the non-isothermal flow through gas-solid
system, several remarks and recommendations are proposed here that need to be
investigated further. The remaining open issues are:
A large number of simulations must be performed in order to capture a wider
range of operating conditions (e.g. high Reynolds number). Then a more elaborate
comparison can be made with well-known heat transfer correlations.
• The results can be validated against experimental results of different systems
such as ordered, random, bidispers and non-spherical systems. Then it can be
investigated experimentally whether one general equation can be used for all
these systems or not.
• Our results prove that the heterogeneity significantly affects the particle HTC.
Characterization of the effects of heterogeneity on heat transfer need careful
numerical or experimental investigations. It would be ideal if the particle HTC
could be characterized according to the local microstructural information.
• The results obtained from fully resolved simulation of a small fluidized bed can
be compared with the results of the same system using coarse-grained methods.
In other words, the gas-solid heat exchange is obtained with coarse-grained mod-
els and then compared to the true value obtained from DNS. This comparison
86 Chapter 7. Summary and recommendations
enables one to assess the performance of coarse-grained methods in simulation
of non-isothermal fluid.
• Natural convection heat transfer is not negligible at high pressure situations.
The effect of natural convection (or buoyancy forces) over the flow pattern and
the heat transfer in a bed can be investigated numerically. These results are
valuable since the experimental option is very expensive and time demanding.
References
A.R. Balakrishnan and D.C.T. Pei. Heat transfer in gas-solid packed bed systems. 3.
overall heat transfer rates in adiabatic beds. Industrial & Engineering Chemistry
Process Design and Development, 18(1):47–50, 1979.
R. Beetstra, M.A. van der Hoef, and J.A.M. Kuipers. Drag force of intermediate
reynolds number flow past mono-and bidisperse arrays of spheres. AIChE Journal,
53(2):489–501, 2007.
S. Benyahia. On the effect of subgrid drag closures. Industrial & Engineering Chem-
istry Research, 49(11):5122–5131, 2009.
R.B. Bird, W.E. Stewart, and E.N. Lightfoot. Transport phenomena. John Wiley &
Sons, 2007.
J.S.M. Botterill. Fluid-bed heat transfer. Academic Press London, New York., 1975.
H.P.A. Calis, J. Nijenhuis, B.C. Paikert, F.M. Dautzenberg, and C.M. Van Den Bleek.
CFD modelling and experimental validation of pressure drop and flow profile in a
novel structured catalytic reactor packing. Chemical Engineering Science, 56(4):
1713–1720, 2001.
A.P. Collier, A.N. Hayhurst, J.L. Richardson, and S.A. Scott. The heat transfer coef-
ficient between a particle and a bed (packed or fluidised) of much larger particles.
Chemical Engineering Science, 59(21):4613–4620, 2004.
D. Darmana, W. Dijkhuizen, N.G. Deen, M. van Sint Annaland, and J.A.M Kuipers.
Detailed 3d modeling of mass transfer processes in two-phase flows with dynamic
interfaces. Proceedings of the 6th international Conference on Multiphase Flow,
2007.
M.L. de Souza-Santos. Solid fuels combustion and gasification: modeling, simulation,
and equipment operations. CRC Press, 2004.
87
88 References
N.G. Deen, S.H.L. Kriebitzsch, van der Hoef, M., and J.A.M. Kuipers. Direct numer-
ical simulation of flow and heat transfer in dense fluid-particle systems. Chemical
Engineering Science, 81:329–344, 2012.
J. Feng, H.H. Hu, and D.D. Joseph. Direct simulation of initial value problems for
the motion of solid bodies in a newtonian fluid. J. Fluid Mech, 261:95–134, 1994.
Z.G. Feng and E.E. Michaelides. A numerical study on the transient heat transfer
from a sphere at high Reynolds and Peclet numbers. International Journal of Heat
and Mass Transfer, 43(2):219–229, 2000.
Z.G. Feng and E.E. Michaelides. Inclusion of heat transfer computations for particle
laden flows. Phys. Fluids, 20:675–684, 2008.
Z.G. Feng and E.E. Michaelides. Heat transfer in particulate flows with direct nu-
merical simulation. Int. J. Heat Mass Transfer, 52:777–786, 2009.
H. Freund, T. Zeiser, F. Huber, E. Klemm, G. Brenner, F. Durst, and G. Emig.
Numerical simulations of single phase reacting flows in randomly packed fixed-bed
reactors and experimental validation. Chemical Engineering Science, 58(3):903–
910, 2003.
D. Goldstein, R. Handler, and L. Sirovich. Modeling a no-slip boundary with an
external force field. J. Comput. Phys, 105:354–366, 1993.
A. Guardo, M. Coussirat, F. Recasens, M.A. Larrayoz, and B. Escaler. CFD study
on particle-to-fluid heat transfer in fixed bed reactors: Convective heat transfer at
lowand high pressure. Chemical Engineering Science, 81:4341–4353, 2006.
D.J. Gunn. Transfer of heat or mass to particles in fixed and fluidised beds. Inter-
national Journal of Heat and Mass Transfer, 21(4):467–476, 1978.
D.J. Gunn and J.F.C. De Souza. Heat transfer and axial dispersion in packed beds.
Chemical Engineering Science, 29(6):1363–1371, 1974.
F.P. Incropera. Introduction to heat transfer. John Wiley & Sons, 2011.
J. Kim and H. Choi. An immersed boundary finite-volume method for simulations
of the heat transfer in complex geometries. Korean Soc. Mech. Eng. Int. J, 18:
1026–1035, 2000.
D. Koch, L. Ladd, and J.C. Anthony. Moderate reynolds number flows through
periodic and random arrays of aligned cylinders. Journal of Fluid Mechanics, 349:
31–66, 1997.
89
S.H.L. Kriebitzsch, M.A. van der Hoef, and J.A.M. Kuipers. Drag force in discrete
particle models-continuum scale or single particle scale? AIChE Journal, 59(1):
316–324, 2013.
D. Kunii and O. Levenspiel. Fluidization engineering. Butterworth-Heinemann,
Boston, 1991.
F. Kuwahara, A. Nakayama, and H. Koyama. A numerical study of thermal dispersion
in porous media. Journal of heat transfer, 118(3):756–761, 1996.
A. Maheshwari, R.P. Chhabra, and G. Biswas. Effect of blockage on drag and heat
transfer from a single sphere and an in-line array of three spheres. Powder Technol,
168:74–83, 2006.
J. Mohd-Yusof. Combined immersed boundaries/b-splines methods for simulations
of flows in complex geometries. Annual Research Briefs, Center for Turbulence
Research, Stanford University, 161:35–60, 1997.
D. Nemec and J. Levec. Flow through packed bed reactors: 1. single-phase flow.
Chemical Engineering Science, 60(24):6947–6957, 2005.
M. Nijemeisland. M.sc. thesis. Worcester Polytechnic Institute. Worcester, MA,
USA., 2000.
E. Nsofor and G.A. Adebiyi. Measurements of the gas-particle convective heat transfer
coefficient in a packed bed for high-temperature energy storage. Experimental
Thermal and Fluid Science, 24(1):1–9, 2001.
M.S. Parmar and A.N. Hayhurst. The heat transfer coefficient for a freely moving
sphere in a bubbling fluidised bed. Chemical Engineering Science, 57(17):3485–
3494, 2002.
C.S. Peskin. Numerical analysis of blood flow in the heart. J. Comput. Phys, 25:
220–252, 1977.
R.S. Ramachandran, C. Kleinstreuer, and T.Y. Wang. Forced convection heat transfer
of interacting spheres. Numerical heat transfer, 15(4):471–487, 1989.
W.E. Ranz and W.R. Marshall. Evaporation from drops. Chem. Eng. Prog, 48:
141–146, 1952.
S.J.P. Romkes, F.M. Dautzenberg, C.M. Van den Bleek, and H.P.A. Calis. CFD
modelling and experimental validation of particle-to-fluid mass and heat transfer in
a packed bed at very low channel to particle diameter ratio. Chemical Engineering
Journal, 96(1):3–13, 2003.
90 References
E.M. Saiki and S. Biringen. Numerical simulation of a cylinder in uniform flow:
application of a virtual boundary method. J. Comput. Phys, 123:450–465, 1996.
T.E.W. Schumann. Heat transfer: a liquid flowing through a porous prism. Journal
of the Franklin Institute, 208(3):405–416, 1929.
S.A. Scott, J.F. Davidson, J.S. Dennis, and A.N Hayhurst. Heat transfer to a single
sphere immersed in beds of particles supplied by gas at rates above and below
minimum fluidization. Industrial & engineering chemistry research, 43(18):5632–
5644, 2004.
T. Shah, M, R.P. Utikar, M.O. Tade, V.K. Pareek, and M.E. Geoffrey. Simulation of
gas–solid flows in riser using energy minimization multiscale model: effect of cluster
diameter correlation. Chemical Engineering Science, 66(14):3291–3300, 2011.
Y. Tang, S.H.L Kriebitzsch, E.A.J.F. Peters, M.A. van der Hoef, and J.A.M Kuipers.
A methodology for highly accurate results of direct numerical simulations: Drag
force in dense gas–solid flows at intermediate reynolds number. International Jour-
nal of Multiphase Flow, 62:73–86, 2014.
H. Tavassoli, S.H.L. Kriebitzsch, M.A. van der Hoef, E.A.J.F. Peters, and J.A.M.
Kuipers. Direct numerical simulation of particulate flow with heat transfer. Inter-
national Journal of Multiphase Flow, 57:29–37, 2013.
S. Tenneti, R. Garg, and S. Subramaniam. Drag law for monodisperse gas–solid
systems using particle-resolved direct numerical simulation of flow past fixed as-
semblies of spheres. International journal of multiphase flow, 37(9):1072–1092,
2011.
S. Tenneti, B. Sun, R. Garg, and S. Subramaniam. Role of fluid heating in dense gas–
solid flow as revealed by particle-resolved direct numerical simulation. International
Journal of Heat and Mass Transfer, 58(1):471–479, 2013.
A.K. Tornberg and B. Engquist. Numerical approximations of singular source terms
in differential equations. Journal of Computational Physics, 200(2):462–488, 2004.
M. Uhlmann. An immersed boundary method with direct forcing for the simulation
of particulate flows. J. Comput. Phys, 209:448–476, 2005.
M.A. van der Hoef, R. Beetstra, and J.A.M. Kuipers. Lattice-Boltzmann simulations
of low-reynolds-number flow past mono-and bidisperse arrays of spheres: results
for the permeability and drag force. Journal of fluid mechanics, 528:233–254, 2005.
M.A. van der Hoef, M. van Sint Annaland, N.G. Deen, and J.A.M Kuipers. Numerical
simulation of dense gas-solid fluidized beds: A multiscale modeling strategy. Annu.
Rev. Fluid Mech., 40:47–70, 2008.
91
N. Wakao, S. Kaguei, and T. Funazkri. Effect of fluid dispersion coefficients on
particle-to-fluid heat transfer coefficients in packed beds: correlation of Nusselt
numbers. Chemical engineering science, 34(3):325–336, 1979.
Z. Wang, J. Fan, K. Luo, and K. Cen. Immersed boundary method for the simulation
of flows with heat transfer. International Journal of Heat and Mass Transfer, 52:
4510–4518, 2009.
S. Whitaker. Forced convection heat transfer correlations for flow in pipes, past
flat plates, single cylinders, single spheres, and for flow in packed beds and tube
bundles. AIChE Journal, 18(2):361–371, 1972.
C.C. Wu and G.J. Hwang. Flow and heat transfer characteristics inside packed and
fluidized beds. Journal of heat transfer, 120(3):667–673, 1998.
J. Yang, Q. Wang, M. Zeng, and A. Nakayama. Computational study of forced
convective heat transfer in structured packed beds with spherical or ellipsoidal
particles. Chemical Engineering Science, 65(2):726–738, 2010.
X. Yin and S. Sundaresan. Fluid-particle drag in low-Reynolds-number polydisperse
gas–solid suspensions. AIChE journal, 55(6):1352–1368, 2009.
List of Publications
Journal Publications
• Tavassoli H. , Kriebitzsch S.H.L., van der Hoef M.A., Peters E.A.J.F., Kuipers
J.A.M., 2013, Direct numerical simulation of particulate flow with heat transfer
, International Journal of Multiphase Flow, 57, 29-37.
• Tavassoli H., Peters E.A.J.F., Kuipers J.A.M., Direct Numerical Simulation of
fluid-particle heat transfer in fixed random arrays of non-spherical particles,
AIChE J, in preparation.
• Tavassoli H., Peters E.A.J.F., Kuipers J.A.M., Characterization of gas-solid
heat exchange by Direct Numerical Simulation , Chem. Eng. Sci., in prepara-
tion.
Publications in Conference Proceedings
• Tavassoli H., Peters E.A.J.F., Kuipers J.A.M., 2013, Direct numerical simu-
lation of the Non-isothermal flow in porous media packed with non-spherical
particles. 8th International Conference on Multiphase Flow (ICMF), Jeju, Ko-
rea.
• Tavassoli H., Peters E.A.J.F., Kuipers J.A.M., 2013, Direct numerical sim-
ulation of fluid-particle heat transfer in dense arrays of non-spherical parti-
cles,. The 14th international conference on fluidization, Noordwijkerhout, The
Netherlands.
Presentations
• Tavassoli H., Peters E.A.J.F., Kuipers J.A.M., 2013, Direct numerical simula-
tion of non-Isothermal flows in a dense gas-solid system. 9th European Congress
of Chemical Engineering,The Hague, The Netherlands.
93
94 References
• Tavassoli H., Peters E.A.J.F., Kuipers J.A.M., 2013, Direct numerical simu-
lation of buoyancy effects on heat transfer in dense fluid-particles Systems.
EUROMECH / ERCOFTAC Colloquium on Immersed Boundary Methods.
• Tavassoli H., Peters E.A.J.F., Kuipers J.A.M., 2014, Direct numerical simula-
tion of non-isothermal flows through stationary arrays of bidisperse spheres, In-
vited presentation at 7th World congress on particle technology, Beijing, China.
Acknowledgements
During the past four years in the Multiphase Reactors Group, Department of Chem-
ical Engineering and Chemistry, Eindhoven University of Technology, I have received
great support from my colleagues, family and friends. Here I would like to express
my sincere thanks to you all for your help, support, discussion and friendship.
First of all, I would like to thank the European Research Council for its financial
support to the project. I am deeply grateful to thanks to my promotor Hans Kuipers
and copromotor Frank Peters who acted as my daily supervisors, for giving me the
opportunity to work on this project and for their dedication in guiding me during
the entire PhD project. They motivated and encouraged me during the research and
writing of this thesis. Their patience, guidance and kind help and discussions in all
aspects have made the past years an ever-good memory in my life. I have learned
a lot from their extensive knowledge and experience in transport phenomena, CFD
and many brilliant and creative ideas both in science and daily life.
My sincere thanks also for Martin van der Hoef, Martin van Sint Annaland and
Niels Deen (Faculty of Multiphase Reactors Group) for the discussions that we had
during the project.
Then I like to thank all other members of the Multiphase Reactors Group, where
I had the privilege to work in an international environment. I would like to thank my
former and current colleagues: Vinayak, Vikrant, Lucia, Yali, Sushil, Sandip, Amit,
Lizzy, Martin, Deepak, Kay, Mohammad, Arash, Mahraz, Lianghui, Mariet, Paul,
Sebastian, Jelle, Ivo for bringing a friendly and creative atmosphere. I wish to extend
my great gratitude to Sebastian who helped me whenever I faced a technical software
and computer problem. I wish to express my thanks to Ada Rijnberg and Judith
Wachters for their help in many administrative matters.
Thanks also to colleagues around the world that have shared their advice or codes
with me, and to the open-source community for making computing a shared resource.
I extend my sincere thanks to my friends and their families in- and outside Eindhoven
(Kazem, Mahmood, Reza, Amin, Mohammad Reza, Parisa, Laleh, Maryam, Cather-
ine, Glenda and Patty) for their friendship and support. All their support made my
stay in the Netherlands possible.
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96 Acknowledgements
Finally, my special gratitude is directed to my parents: for giving me their deep
love, sacrifice, encouragement and support during my years of education. Also for my
brothers: Masoud and Saeed. This thesis is dedicated to them.
Hamid Tavassoli
Eindhoven, November, 2014
Curriculum Vitae
Hamid Tavassoli was born on 15-9-1982 in Shiraz, Iran. After completing his sec-
ondary education at Shiraz in 2001, he started his chemical engineering study at
Shiraz University. Then he started his master program (2004-2006) at the Sharif
University of Technology in Tehran, Iran. He defended his graduation project on
simulation of chemical reaction (the isomerization of Glucose-Fructose) with mobi-
lized catalyst in the reactor.
From 2007, he worked as a consultant engineer at the Research Institute of
Petroleum Industry, Iran, to perform research on the Gas to Liquids (GTL) tech-
nology. Afterwards, he started his PhD research in April 2010 at the Multi-scale
Modeling of Multi-phase Flows group at chemical engineering department of Eind-
hoven University, the Netherlands, supervised by his promotor Prof. J.A.M. Kuipers
and co-promotor Dr. E.A.J.F. Peters. The results of this research are presented in
this thesis.
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