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

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

n−1

)

Δt

, withΔt the time step, (4)

according to the following schemes:

• Euler: f (u

n+1

, u

n

, u

n−1

) = u

n+1

− u

n

,

• Gear: f (u

n+1

, u

n

, u

n−1

) =

3

2

u

n+1

− 2u

n

+

1

2

u

n−1

.

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

n−1

) ·∇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

n−1

)

Δt

+ ((u

n+1,k−1

·∇)u

n+1,k

−∇(r∇·u

n+1,k

) =−∇p

n,k−1

+ ρg

+∇·

μ

∇u

n+1,k

+ (∇

T

u

n+1,k

)

+ F

n+1,k

si

,

p

n+1,k

= p

n,k−1

−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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