The extended discrete element method (XDEM) applied to drying of a packed bed

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The Extended Discrete Element Method

(XDEM) Applied to Drying of a Packed Bed

B. Peters, X. Besseron, A. Estupinan, F. Hoffmann, M. Michael, A. Mahmoudi

Université du Luxembourg, Faculté des Sciences, de la Technologie et de la Communication 6, rue Coudenhove-Kalergi, L-1359 Luxembourg

Corresponding author email: bernhard.peters@uni.lu Phone: (+352) 46 66 44 5496 Fax: (+352) 46 66 44 52 00

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Abstract

A vast number of engineering applications involve physics not solely of a single domain but of several physical phenomena, and therefore are referred to as multi-physical. As long as the phenomena considered are to be treated by either a continuous (i.e. Eulerian) or discrete (i.e. Lagrangian) approach, numerical solution methods may be employed to solve the problem. However, numerous challenges in engineering exist and evolve; those include modelling a continuous and discrete phase simultaneously, which cannot be solved accurately by continuous or discrete approaches only. Problems that involve both a continuous and a discrete phase are important in applications as diverse as the pharmaceutical industry, the food processing industry, mining, construction, agricultural machinery, metals manufacturing, energy production and systems biology. A novel technique referred to as Extended Discrete Element Method (XDEM) has been developed that offers a significant advancement for coupled discrete and continuous numerical simulation concepts. XDEM extends the dynamics of granular materials or particles as described through the classical discrete element method (DEM) to include additional properties such as the thermodynamic state or stress/strain for each particle coupled to a continuous phase such as a fluid flow or a solid structure. Contrary to a continuum mechanics concept, XDEM aims at resolving the particulate phase through the various processes attached to particles. While DEM predicts the spatial-temporal position and orientation for each particle, XDEM additionally estimates properties such as the internal temperature and/or species distribution during drying, pyrolysis or combustion of solid fuel material such as biomass in a packed bed. These predictive capabilities are further extended by an interaction with fluid flow by heat, mass and momentum transfer and the impact of particles on structures.

Keywords: Extended Discrete Element Method, process engineering, multi-physics, modelling, drying, packed bed.

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1 Introduction

Numerical approaches to modelling multi-phase flow phenomena including a solid (e.g. particulates) may be classified into two categories. The first treats all phases as continua on a macroscopic level, of which the two-fluid model is the most well-known representative [1]. It is well suited to process modelling due to its computational convenience and efficiency. However, all data concerning size distribution, shape or material properties of individual particles is lost to a large extent in an Eulerian formulation due to the averaging concept, as described in detail by Faghri et al. [2]. Therefore, this loss of information on small scales has to be compensated for by additional constitutive or closure relations. Although the Method of Moments (MOM) with is numerical derivatives such as the Quadrature Method of Moments (QMOM) or Direct Quadrature Method of Moments (DQMOM) [3] could be employed to estimate properties of the solid phase, an analytical method based on conservation equations for mass and energy for the particles is believed to be more accurate, avoiding the statistical uncertainty of the Method of Moments.

The second approach considers the solid phase as discrete, while the flow of liquids or gases is treated as a continuous phase. First developments date back to the early 1950s, when Harlow [4] developed the particle-in-cell (PIC) method to solve fluid mechanics problems. In this method, individual particles or fluid elements are tracked by a Lagrangian framework, whereas moments of the distribution, such as densities, are predicted simultaneously in an Eulerian frame of reference.

The Material Point Method (MPM) is one of the latest developments in PIC methods. Initially, PIC methods suffered from excessive energy dissipation, which was resolved by the Fluid Implicit Particle (FLIP) method introduced by Brackbill and Ruppel [5]. The FLIP technique was modified and tailored for applications in solid mechanics by Sulsky et al. [6, 7] and has since been referred to as the Material Point Method [8].

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A further recent development is the Combined Continuum and Discrete Model (CCDM) [9, 10, 11, 12], that also couples a Lagragian with an Eulerian technique. Its application covers most process engineering scenarios of packed, moving and fluidised beds in chemical reactors. In such scenarios the solid phase, consisting of individual particles, is treated discretely, whereas the flow of gases or liquids in the void space is described by Eulerian equations. The discrete description of the solid phase means constitutive relations are omitted and leads to a better understanding of the fundamentals. This was also concluded by Zhu et al. [13] and Zhu et al. [14] during a review of particulate flows modelled with the CCDM approach. It has been a major development in the last two decades and describes the motion of the solid phase by the Discrete Element Method (DEM) on an individual particle scale; the remaining phases are treated by the Navier-Stokes equations. The method is recognized as an effective tool for investigating the interaction between particulate and fluid phases as reviewed by Yu and Xu [15], Feng and Yu [16], and Deen et al. [17].

2 Extended Discrete Element Method (XDEM)

Contrary to the continuum mechanics concept, the Extended Discrete Element Method (XDEM) aims at resolving the particulate phase with its various processes attached to the particles. While methods such as the Method of Moments (MOM), the Material Point Method (MPM) or the Combined Continuum and Discrete Model (CCDM) largely deal with fluid dynamic aspects of multiphase flow, XDEM is a numerical technique that extends the properties of solid particles using a detailed thermodynamic description including chemical reactions, and is based on conservation equations for mass, energy and momentum. The Discrete Element Method is employed to predict position and orientation in space and time for each particle and the Extended Discrete Element Method additionally estimates properties such as internal temperature and/or particle distribution, or mechanical impact with structures. Relevant areas of application include furnaces for wood combustion, blast furnaces for steel production, fluidized beds, cement kilns, and predictions of emissions from combustion of coal or biomass.

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The Extended Discrete Element Method considers each particle of an ensemble as an individual entity with motion and thermodynamics attached to it as described in detail by Peters [18]. The motion of particles is described by a sufficient number of geometric shapes that are believed to cover a large range of engineering applications, while the thermodynamics of a particle incorporates a physical-chemical approach that describes temperature and arbitrary reaction processes for each particle in the ensemble. The exchange of data between continuous and discrete solutions requires careful coordination and a complex feed-back loop so that the coupled analysis converges to an accurate solution. This is performed by coupling algorithms between the Discrete Particle Method to the Finite Volume, using, for example, Computational Fluid Dynamics (CFD).

2.1 Numerical Approach to Motion of Particles

The Discrete Element Method (DEM), also called a Distinct Element Method, is probably the most often applied numerical approach to describe the trajectories of all particles in a system. DEM is a widely accepted and effective method to address engineering problems in granular and discontinuous materials, especially in granular flows, rock mechanics, and powder mechanics. Pioneering work in this domain has been carried out by Cundall [19], Haff [20], Herrmann [21] and Walton [22]. For a more detailed review the reader is referred to Peters [23].

2.2 Numerical Approach to Thermodynamics of Particles

In these numerical approaches to the thermodynamics of particles, an individual particle is considered to consist of a gas, liquid, solid and inert phase, with the inert, solid and liquid species being considered immobile. The gas phase represents the porous structure of a particle and is assumed to behave as an ideal gas. Each of the phases may undergo various conversions by homogeneous, heterogeneous or intrinsic reactions whereby the products may experience a phase change such as encountered during drying. Furthermore, local thermal equilibrium between the phases is assumed, based on the assessment of the ratio of heat transfer by conduction to the rate of heat transfer by convection expressed by the Peclet number as described by Peters [24] and Kansa et al.

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[25]. The conservations of mass, momentum and energy are described by transient and one-dimensional differential conservation equations. In general, the inertial term of the momentum equation is negligible due to a small pore diameter and a low Reynolds number. Thermal conversion of particles is predicted by relevant and validated kinetic data that allows prediction of both temperature distribution and chemical reactions for an individual particle. This concept is applied to each particle within the packed bed, for which spatial and temporal distributions are resolved accurately.

2.3 Computational Fluid Dynamics (CFD) of Packed Beds

Packed beds can be characterised as a type of porous media in which fluid flow behaves more like an external flow. The flow may be accurately described for a continuum approach by averaging relevant variables and parameters on a coarser level. This leads to a formulation where the actual multiphase medium, consisting of solid matrix and fluid, is treated as a flow though a porous media for which the transient and 3-dimensional differential conservation equations for mass, momentum and energy are solved. The current approach has the advantage that the distribution of particles and their volumes are known by predictions of the motion module, so that the distribution of porosity within the flow field, in particular near walls, is readily available. Therefore, no further correlations for porosity distributions are required. This feature of the current approach leads to an accurate prediction of velocity and temperature distributions of the flow field, of which the temperature and composition of the gas in the vicinity of the particles determine heat and mass transfer by appropriate transfer coefficients.

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3 Results and Discussion

The following results present predictions of the flow behaviour, temperature distribution and drying process in a packed bed, as well as validation of the predictions.

3.1 Flow Characteristics of a Randomly Packed Bed

A classical continuous representation of particulate matter requires either experimental data or empirical correlations to determine both the total surface area of the particles and the distribution of void space between them. These disadvantages are avoided using the current approach. XDEM evaluates the available surface for heat transfer and porosity (i.e. void space) influencing the flow distribution. Of particular interest is the distribution of void space in near-wall regions and its effect on flow, heat and mass transfer.

A reactor was randomly filled with particles and the final arrangement allowed the assessment of local heat and mass transfer conditions. In particular, the distribution of velocity and porosity were as shown in Fig. 1.

Figure 1: Distribution of porosity and axial gas velocity in a cross-sectional area of a randomly packed bed

An important characteristic of packed beds is the wall effect; this is manifested by an increased void space along the inner walls. At certain positions around the wall the void space reaches values of 0.43, higher than the inner regions, where the void space

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decreases to 0.3. In addition, the void space does not show any regular pattern, but is distributed rather inhomogeneously, dependent on the packaging of the particles. Because of this, the fluid flow experiences less drag, resulting in an increased mass flow rate along the walls, with a velocity of 0.67 m/s, as depicted in Fig. 1. It contributes to an increased heat transfer to near-wall particles, and therefore to a higher drying rate. Furthermore, a higher heat transfer to the wall may cause increased thermal losses of the entire reactor.

3.2 Drying of a Randomly Packed Bed

Similar to the set-up presented in the previous section, the drying of wood particles was predicted and compared to experimental results. Measurements were carried out with a cylindrical reactor that was filled with approximately 3 kg of biomass. The packed bed was heated by a hot stream of nitrogen and the mass loss due to drying was recorded by weight measurements. For a detailed description of the experimental facility and measurements the reader is referred to Peters [18].

The drying process is described by a heat sink model [26], in contrast to the kinetic approaches employed by Chan et al. [27] and Krieger-Brockett and Glaister [28]. It is believed that a balance between available energy for evaporation may offer a more universal approach, as expressed by:

̇ {

( )

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where T, Tevap, Hevap, ρ and cp denote the local particle temperature, the evaporation

temperature, the evaporation enthalpy, the density and the specific heat of the particle. Fig. 2 shows the results of the drying process in the form of an integral loss of moisture over the drying period of approximately 160 minutes for two temperatures of the incoming gas (T = 408 K and T = 423 K). After an initial heating period of approximately 1000 s for the packed bed, during which heat is transferred from the gas

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to the particles, evaporation conditions are met. At this point, some particles have already reached the evaporation temperature, meaning water vapour has been produced inside the particle and successively transported into the gas phase. As time progresses, more and more particles are effected in this way, so that the total weight of the packed bed is reduced. After periods of approximately 7000 s and 11000 s, the packed bed is completely dried at temperatures of 423 K and 403 K, respectively. A very good agreement between measurements and predictions was achieved, as depicted in Fig. 2.

Figure 2: Comparison between measurements and predictions for drying of a randomly packed bed

Since the XDEM methodology resolves individual particles in conjunction with the gas phase, details of the underlying physics are revealed. These detailed results are depicted in Fig. 3a-3f at different instances of time, for which both the water content of the particles and the gas temperature is shown. During the initial period of 1000 s, as shown in Figs. 2 and 3a, no drying takes place, because the incoming hot gas is heating the particles to the evaporation temperature. After 2000 s, the first particles have reached

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the evaporation temperature and consequently reduce their water content. As seen by the distribution of the water content of individual particles in Fig. 3b, the drying process proceeds rather heterogeneously across the top-down cross-section of the packed bed. This is due to the inhomogeneously distributed particles that affect both the flow distribution and heat transfer to the gas phase and between particles in contact. This rather heterogeneous drying pattern progresses through the packed bed as shown in the following figures of Fig. 3c-3f and is not comparable to a drying front propagating through a bed as often assumed by many authors [29].

Figure 3a: Distribution of gas temperature and moisture content of individual particles at time t = 1000 s

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Figure 3b: Distribution of gas temperature and moisture content of individual particles at time t = 2000 s

Figure 3c: Distribution of gas temperature and moisture content of individual particles at time t = 3000 s

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Figure 3d: Distribution of gas temperature and moisture content of individual particles at time t = 4000 s

Figure 3e: Distribution of gas temperature and moisture content of individual particles at time t = 5000 s

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Figure 3f: Distribution of gas temperature and moisture content of individual particles at time t = 6000 s

4 Summary

The current contribution introduces the numerical concept of the Extended Discrete Element Method (XDEM) to describe the thermal conversion of packed beds using the example process of the drying of wood particles. One of the key features is that the solid phase (consisting of wood particles) is discretely resolved. It considers each particle as an individual entity for which one-dimensional and transient conservation equations for mass, momentum and energy are solved. Each particle is in contact through heat and mass transfer between its surface and the surrounding gas phase. The latter is described by classical approaches of computational fluid dynamics for gaseous flow in porous media. Hence the Extended Discrete Element Method is applicable to a large variety of thermal conversion processes, as demonstrated by the example in this paper of drying of a packed bed of wood. The predicted results represent distributions in time and space for major properties such as temperature and species for each individual particle, and therefore give process detail at the smallest scale which cannot be obtained by measurement. Hence, analysing this data contributes significantly to the knowledge of

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the underlying physics of packed beds. The drying process is considered to be an important process before pyrolysis and combustion take place. The latter processes are predicted similarly by the same concept, using kinetic models describe to pyrolysis [30, 31] and combustion [32].

5 Acknowledgements

We would like to acknowledge gratefully the contribution of the Fonds National de la Recherche, Luxembourg, for funding this work.

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