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Direct numerical simulation of dense gas-solid non-isothermal flowsTavassoli Estahbanati, H.

DOI:10.6100/IR782478

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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).


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


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).


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.


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)


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.


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)


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)


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


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.


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.


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,


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


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)


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


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.


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


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


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


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.


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


(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).


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


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.


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)


6.1 Local HTC




Tf,loc(~xp) =

∫∫∫Vf (~xp)


U(~x)dVf. (19)


hp,loc =Qp

Ap ∆Tloc. (20)

12

(b)

275

290

305

320

(a)

275

290

305

320

(b)

0

2

4

6

(c)


6.1 Local HTC




Tf,loc(~xp) =

∫∫∫Vf (~xp)


U(~x)dVf. (19)


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 .


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


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


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


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


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


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.


(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.


(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.


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


Figure 5.5: Particle Nusselt number Nu, NuL, NuS when φ=0.4.




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


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


5

(a)

0 5 10 15 20 250

0.1

0.2

0.3

Nup

Probabilitydensity

function

Nuip,SNuip,L


0 5 10 15 200

0.1

0.2

0.3

Nup

Probab

ilitydensity

function

Nuip,SNuip,L


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.


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




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


(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.


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.


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)


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


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.


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.


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.

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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.

95

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.

97

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