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Journal ArticleDOI

Particle-resolved numerical simulations of the gas–solid heat transfer in arrays of random motionless particles

05 Feb 2019-Acta Mechanica (Springer Vienna)-Vol. 230, Iss: 2, pp 541-567
TL;DR: In this paper, particle-resolved direct numerical simulations of non-isothermal gas-solid flows have been performed and analyzed from microscopic to macroscopic scales using a Lagrangian VOF approach based on fictitious domain framework and penalty methods.
Abstract: Particle-resolved direct numerical simulations of non-isothermal gas–solid flows have been performed and analyzed from microscopic to macroscopic scales. The numerical configuration consists in an assembly of random motionless spherical particles exchanging heat with the surrounding moving fluid throughout the solid surface. Numerical simulations have been carried out using a Lagrangian VOF approach based on fictitious domain framework and penalty methods. The entire numerical approach (numerical solution and post-processing) has first been validated on a single particle through academic test cases of heat transfer by pure diffusion and by forced convection for which analytical solution or empirical correlations are available from the literature. Then, it has been used for simulating gas–solid heat exchanges in dense regimes, fully resolving fluid velocity and temperature evolving within random arrays of fixed particles. Three Reynolds numbers and four solid volume fractions, for unity Prandtl number, have been investigated. Two Nusselt numbers based, respectively, on the fluid temperature and on the bulk (cup-mixing) temperature have been computed and analyzed. Numerical results revealed differences between the two Nusselt numbers for a selected operating point. This outcome shows the inadequacy of the Nusselt number based on the bulk temperature to accurately reproduce the heat transfer rate when an Eulerian–Eulerian approach is used. Finally, a connection between the ratio of the two Nusselt numbers and the fluctuating fluid velocity–temperature correlation in the mean flow direction is pointed out. Based on such a Nusselt number ratio, a model is proposed for it.

Summary (1 min read)

1 Introduction

  • This study deals with the analysis and the modeling of the heat transfer in dense particle-laden flows.
  • High Reynolds and Prandtl numbers are instead difficult to reproduce because of the small boundary layer thickness and therefore the requirement of even more refined grids.

2 Numerical modeling

  • 1 Governing equations and solution methods A Lagrangian VOF approach using fictitious domains and penalty methods is used in the present work.
  • These authors considered the solid particle phase as a continuous phase with high viscosity, requiring a treatment of discontinuities especially for density and viscosity at the interface.
  • The temporal derivatives are approximated with implicit finite volume schemes which does not require a stability condition; either Euler or Gear schemes are used depending on the complexity of the problem.
  • Linked to the previous algebraic parameter, solid constraints are ensured at the same time as incompressibility with second-order convergence in space.
  • 2 Numerically, the computation of Qp→f needs a discretization of the sphere surface.

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To cite this version:
Thiam, Elhadji Ibrahima and Masi, Enrica and Climent, Éric
and Simonin, Olivier and Vincent, Stéphane Particle-resolved
numerical simulations of the gas–solid heat transfer in arrays
of random motionless particles. (2019) Acta Mechanica, 230.
541-567. ISSN 0001-5970.
Official URL:
https://doi.org/10.1007/s00707-018-2346-5
Open Archive Toulouse Archive Ouverte

Elhadji I. Thiam · Enrica Masi · Eric Climent ·
Olivier Simonin · Stéphane Vincent
Particle-resolved numerical simulations of the gas–solid heat
transfer in arrays of random motionless particles
Abstract Particle-resolved direct numerical simulations of non-isothermal gas–solid flows have been per-
formed and analyzed from microscopic to macroscopic scales. The numerical configuration consists in an
assembly of random motionless spherical particles exchanging
heat with the surrounding moving fluid
through-out the solid surface. Numerical simulations have been carried out using a Lagrangian VOF
approach based on fictitious domain framework and penalty methods. The entire numerical approach
(numerical solution and post-processing) has first been validated on a single particle through academic test
cases of heat transfer by pure diffusion and by forced convection for which analytical solution or empirical
correlations are available from the literature. Then, it has been used for simulating gas–solid heat exchanges
in dense regimes, fully resolving fluid velocity and temperature evolving within random arrays of fixed
particles. Three Reynolds numbers and four solid volume fractions, for unity Prandtl number, have been
investigated. Two Nusselt numbers based, respectively, on the fluid temperature and on the bulk (cup-
mixing) temperature have been computed and analyzed. Numerical results revealed differences between the
two Nusselt numbers for a selected operating point. This outcome shows the inadequacy of the Nusselt
number based on the bulk temperature to accurately reproduce the heat transfer rate when an Eulerian–
Eulerian approach is used. Finally, a connection between the ratio of the two Nusselt numbers and the
fluctuating fluid velocity–temperature correlation in the mean flow direction is pointed out. Based on such a
Nusselt number ratio, a model is proposed for it.
1 Introduction
This study deals with the analysis and the modeling of the heat transfer in dense particle-laden flows. Such
a regime covers a wide spectrum of industrial applications dealing with energy conversion, manufacturing
processes, waste recycling, etc. Many of these applications need to recast their processes in order to comply with
ne w energy and climate targets, thus increasing ef ficiency while reducing g as emissions. Most of them involve
reacti ve flows in which the heat e xchanged between the solid and gaseous phases, and between each phase and
the wall, plays a crucial role in the entire process. An understanding of the heat transfers in such complex flows,
a long-standing issue, is therefore essential to be able to enhance the performances of existing processes and
the development of new technologies. Accordingly, gas–solid heat exchanges have been extensively studied
over the years. The particle to fluid heat transfer coefficient in dense regimes (typically fixed or fluidized
beds) has been evaluated under theoretical and experimental studies. In the experiments, various methods,
E. I. Thiam · E. Masi (
B
) · E. Climent · O. Simonin
Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS, Toulouse, France
E-mail: enrica.masi@imft.fr
S. Vincent
Université Paris-Est Marne la Vallée, Laboratoire de Modélisation et Simulation Multi Echelle (MSME),
UMR 8208 Champs-Sur-Marne, France

designs and operating conditions have been used to determine the heat transfer coefficient over a large range
of operating points [1]. However, experimental results exhibited a somewhat large disparity to each other
which may be attributed to different experimental techniques employed or, as suggested by Gunn [2], to the
different interpretations of raw data. The heat transfer coefficient is indeed the result of a model applied to the
experimental quantities, and it strongly depends on the assumptions made. For example, it has been shown
and extensively discussed that accounting or not for the axial dispersion in the modeling substantially affects
the estimation of the Nusselt number at low Reynolds numbers [1,2]. Moreover, experimental investigations
cannot provide a local view of the flow behavior and a deep understanding of the related microscopic features.
To overcomethese limits, numerical simulationmay be used. Thelatter represents indeed apowerfulalternative
to experimental investigations, as it is a non-intrusive method able to fully access the local quantities of the
particulate flows. To allow the numerical simulation to provide trustworthy heat transfer coefficients, a high
accuracy of the results has to be ensured. A high level of accuracy is subject to high resolution, which implies
very fine meshes and consequently high computational costs. With the development of high-performance
computing (HPC), the direct numerical simulation at microscopic scale (that is at a scale comparable to the
particle dimensions) is becoming affordable and thus usable for the investigation of heat exchanges in dense
suspensions. By the numerical simulation, Reynolds and Prandtl numbers may be easily changed over a range
of intermediate values, thus making it possible to provide Nusselt number correlations as a function of the
solid volume fraction and the two aforementioned dimensionless groups. High Reynolds and Prandtl numbers
are instead difficult to reproduce because of the small boundary layer thickness and therefore the requirement
of even more refined grids. In these last years, several studies using the direct numerical simulation (also
referred to as fully resolved or particle-resolved DNS) have been carried out in order to investigate the heat
transfer in dense regimes, over intermediate Reynolds and Prandtl numbers and solid concentration up to 50%.
These studies employed different numerical strategies for solving the flow interacting with the solid bodies.
For example, an immersed boundary method (IBM) for non-isothermal particulate flows was used by Feng
and Michaelides [3], Deen et al. [4] and Feng and Musong [5]. Tavassoli et al. [6] extended the approach
originally proposed by Uhlmann [7] to account for the heat transport in order to study the heat transfer in
particulate flows. These authors reported numerically assessed Nusselt numbers in a random array of fixed
spheres in which the fluid flows from an inlet boundary toward an outlet boundary exchanging heat with the
solid phase. They provided comparisons with the well-known Gunn correlation [2] and pointed out deviations
increasing with the solid volume fractions, considered consistent with the accuracy of such a correlation.
Deen et al. [8] reviewed the DNS methods and on the basis of available data refit the Gunn correlation and
thus provided a new correlation. The particle-resolved uncontaminated-fluid reconcilable immersed boundary
method (PUReIBM) was extended and used in non-isothermal conditions by Tenneti et al. [9] to perform direct
numerical simulations of gas–solid heat exchanges within an assembly of random spheres, by using a fully
periodic configuration based on a thermal similarity boundary condition for the temperature. Sun et al. [10]
suggested a new correlation for the Nusselt number as well as a correction factor to be used in the frame of
an Eulerian–Eulerian formulation. Kruggel-Eemden et al. [11] used a lattice Boltzmann method (LBM) to
investigate gas-particle heat transfers. Periodic boundary conditions for the flow together with constant and
adiabatic conditions at the streamwise boundaries for the temperature were used to simulate heat exchanged
in the assembly of random particles. Including the axial dispersion, by using the axial dispersion coefficient
proposed by Wakao [1], they obtained Nusselt numbers in good agreement with the correlation proposed
by Tavassoli et al. [12]. A new method combining immersed boundary and fictitious domain (referred to as
HFD-IB) was recently developed and used to investigate the heat transfer in bi-dispersed regimes by Municchi
and Radl [13]. Focusing on the Euler–Lagrange approaches for particulate flows, these authors proposed a
closure for the particle Nusselt number as a function of the particle drag force. Alternative methods are also
emerging–see, for example, the PHYSALIS method extended to non-isothermal particulate flows by Wang
et al. [14]. In the present work, a Lagrangian VOF approach using fictitious domains and penalty methods
[15] is used to perform particle-resolved numerical simulations of gas–solid heat transfers. In Sect. 2,such
an approach is briefly recalled. A preliminary study devoted to validate the entire methodology (including
post-processing strategies) is described in Sect. 3. Direct numerical simulations of gas–solid heat exchanges
in arrays of random motionless particles are finally presented in Sect. 4. In the latter, numerical results on two
Nusselt numbers based, respectively, on the fluid temperature and on the bulk (cup-mixing) temperature are
presented and discussed. Finally, a connection between the ratio of such Nusselt numbers and the fluctuating
fluid velocity–temperature term appearing in the energy conservation equation is pointed out. On the basis of
this Nusselt number ratio, a model is proposed for it.

2 Numerical modeling
2.1 Governing equations and solution methods
A Lagrangian VOF approach using fictitious domains and penalty methods is used in the present work. It is
based on an Eulerian formulation of the Navier–Stokes equations discretized on a fixed structured grid. This
approachwasinitiatedbyRitzandCaltagirone[16]forhandlingparticulateflows.Tomodelthebehavioroffluid
and solid phases, the one-fluid model of Kataoka [17], initially devoted to deformable interfaces and fluid/fluid
two-phase flows, was extended to flows interacting with moving finite-size particles by Ritz and Caltagirone
[16]. These authors considered the solid particle phase as a continuous phase with high viscosity, requiring
a treatment of discontinuities especially for density and viscosity at the interface. With an arithmetic average
for the density and a harmonic average for the viscosity at the fluid–solid interfaces, the Stokes flow around a
circular cylinder and two-dimensional sedimentation of particles were simulated [16]. This methodology has
undergone several improvements, and now, its originality comes from the reformulation of the stress tensor
μ
u + (
T
u)

as proposed by Caltagirone and Vincent [18]. It consists of a decomposition of the stress
tensor for Newtonian fluids in order to distinguish the contributions of tearing, shearing and rotation. With the
help of a phase function C (= 0influidmediumand= 1 in solid medium), which describes the solid phase
shape evolution through an advection equation (Eq. (2)), classical Navier–Stokes equations are solved for both
phases, taking into account the phase behavior:
∇·u = 0,
ρ
u
t
+ (u ·∇)u
=−p +∇·
μ
u + (
T
u)

+ ρg + F
si
. (1)
In the above system, u = (
u ,
v,
w) and g are, respectively, the velocity and the gravity vectors, p is the
pressure field, ρ and μ are the density and the dynamic viscosity and F
si
is the force ensuring coupling
between the phases. The spatial and temporal evolution of the phase function then writes:
C
t
+ u ·∇C = 0. (2)
Equation (2) is solved in a Lagrangian manner. The shape of the particles is tracked by a Lagrangian mesh
made of triangles in 3D. For spherical particles as in the present work, the advection of the solid phase is
satisfied with the Lagrangian tracking of the barycenter of the sphere, using a Runge–Kutta method of second
order. The Eulerian phase function is finally obtained at each time step by projecting the Lagrangian meshes of
all particles on the Eulerian grid with a kind of Monte Carlo approach. All these procedures are detailed [15].
According to the penalty method acting on the viscosity, no tearing, no shearing and constant rotation could
be imposed, for example, to the solid phase. By this approach, the divergence of the viscous stress tensor is
indeed written using the decomposition
∇·
μ
u + (
T
u)

=∇·
[
κ(u)
]
+∇·
[
ζ (u)
]
−∇·
[
η(u)
]
, (3)
which makes easier the implementation of a penalty method by imposing separate viscosity coefficients such as
the tearing viscosity, κ, the shearing viscosity, ζ , and the rotation viscosity, η, appearing in Eq. (3). The implicit
tensorial penalty method (ITPM) for solid behavior and incompressibility constraint is a new evolution, of
second-order convergence in space, of the viscous penalty method. Details about this method may be found in
Vincentetal.[15]. It is implemented together with an augmented Lagrangian method first proposed by Fortin
and Glowinski [19]. Before explaining the specificity of ITPM, we recall the time discretization employed
for solving the Eulerian system (1). The temporal derivatives are approximated with implicit finite volume
schemes which does not require a stability condition; either Euler or Gear schemes are used depending on the
complexityof theproblem. A second-order centeredscheme is employedto approximate the spatialderivatives.
Time derivatives may be written as
u
t
f (u
n+1
, u
n
, u
n1
)
Δt
, withΔt the time step, (4)
according to the following schemes:

Euler: f (u
n+1
, u
n
, u
n1
) = u
n+1
u
n
,
Gear: f (u
n+1
, u
n
, u
n1
) =
3
2
u
n+1
2u
n
+
1
2
u
n1
.
If the Gear scheme is used, the inertial term is linearized by an Adams–Bashforth scheme as follows:
u
n+1
·∇u
n+1
(2u
n
u
n1
) ·∇u
n+1
. The augmented Lagrangian method is used to satisfy the incom-
pressibility constraint through a velocity–pressure (u, p) coupling, by solving a minimization problem. The
approximation of the solution by an Uzawa-like scheme reads:
while ∇ · u
n+1,k
>
1
ρ
f (u
n+1
,u
n
,u
n1
)
Δt
+ ((u
n+1,k1
·∇)u
n+1,k
−∇(r∇·u
n+1,k
) =−p
n,k1
+ ρg
+∇·
μ
u
n+1,k
+ (
T
u
n+1,k
)

+ F
n+1,k
si
,
p
n+1,k
= p
n,k1
r∇·u
n+1,k
.
(5)
In the above system, k is the iterative index for the Uzawa optimization algorithm and n the physical time
iterative index. The significant parameter in Eq. (5) is the augmented Lagrangian parameter r. In the standard
form of the algorithm, r is constant; improvements proposed in [20] used instead a spatial and time parameter
r(x, y, z, t) linked to a fixed initial a priori constant value to get a satisfactory solution. Further improvements
byVincentetal.[21] proved that an algebraic parameter r is suitable to fully carry out incompressibility and
solid constraints in an optimal way. This algebraic parameter is defined according to the discretization matrix
containing the viscous penalty contributions. To implement the penalty method for the viscosity, thanks to the
viscous stress tensor decomposition (3), and in order to impose no shearing, no tearing and constant rotation
for solid particles, a dual grid (points located at the center of the grid cells) is introduced [15]; the latter
allows the specification of shearing and rotation viscosities, while the elongation viscosity is defined on the
pressure nodes. Linked to the previous algebraic parameter, solid constraints are ensured at the same time as
incompressibility with second-order convergence in space. Then, physical properties at fluid–solid interfaces
are defined by using a harmonic average for the viscosities and an arithmetic average for the density. The
particle interaction force F
si
accounting for particle–particle and particle–wall collisions was implemented
and validated by Brändle de Motta et al. [22]. Details about particle tracking and four-way coupling may be
found elsewhere [15].
Whenthe particle velocitiesare nota priori known,ITPM makes it possibleto ensure both incompressibility
and solid constraints, while, for fixed particles, a simpler penalty method may be employed. The latter, referred
to as Darcy penalty method (DPM) [23], is an approach typically used in porous media in order to solve the
Navier–Stokes equations accounting for the interactions with a solid object. It consists in considering an
additional term in the momentum equation based on a local permeability parameter:
ρ
u
t
+ u ·∇u
+
μ
K
u =∇·
μ
u +∇
T
u

−∇p + ρg. (6)
The permeability K tends to +∞in the fluid medium and to zero in the solid medium. This method is employed
to impose a zero velocity inside the solid. Similarly, a constant temperature can be imposed to the solid. In the
energy conservation equation,
ρC
p
T
t
+ u ·∇T
+ β(T T
s
) =∇·[k
f
T ], (7)
where T is the phase temperature (with T
s
the solid one). C
p
is the mass heat capacity and k
f
is the thermal
conductivity; their respective values are set equal for both the phases in this work. The supplementary term
β(T T
s
) is only active in those zones in which the phase function is equal to unity (C = 1) and β →+.
In the fluid domain, C = 0andβ = 0. The finite volume discretization scheme for the energy conservation
equation is based on an explicit total variation diminishing (TVD) scheme for the convection terms, while
an implicit centered scheme is used for the conductive terms. An implicit Euler time discretization is used
for time derivatives. Linear systems resulting from all discretizations (augmented Lagrangian terms, Navier–
Stokes equations, energy equation) are treated with a BiCGSTAB II solver and a modified and incomplete LU
preconditioner [15].

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Frequently Asked Questions (1)
Q1. What are the contributions mentioned in the paper "Particle-resolved numerical simulations of the gas–solid heat transfer in arrays of random motionless particles" ?

The numerical configuration consists in an assembly of random motionless spherical particles exchanging heat with the surrounding moving fluid through-out the solid surface. The entire numerical approach ( numerical solution and post-processing ) has first been validated on a single particle through academic test cases of heat transfer by pure diffusion and by forced convection for which analytical solution or empirical correlations are available from the literature.