Application of Lattice Boltzmann method to meso

Chemical Engineering Science 172 (2017) 503–512

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Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces

Application of Lattice Boltzmann method to meso-scale modelling of coal devolatilisation Arkadiusz Grucelski ⇑, Jacek Pozorski ´ sk, Poland Institute of Fluid Flow Machinery, Polish Academy of Sciences, ul. Fiszera 14, 80-231 Gdan

h i g h l i g h t s Coal devolatilisation modelling at the single grain scale in representative element of volume. Simplification of the Functional Group method for meso-scale modelling. Application of Lattice Boltzmann method for flow, heat transfer and chemical reactions in 3D.

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 20 February 2017 Received in revised form 9 May 2017 Accepted 8 June 2017 Available online 19 June 2017

Thermomechanics of the flow through a granular medium with subsequent release of chemical compounds from the grains of solid fuel is addressed. The modified pyrolysis model, based on the functional groups, is developed for investigation of the devolatilisation process of coal particles. The Lattice Boltzmann method (LBM) is applied with additional distribution functions used to model the evolution of temperature and chemical species. First, as a validation case, the gas and tar release rates from a single grain are found to compare reasonably with experimental data and other models known from the literature. Then, the reactive granular medium, composed of coal grains, is described at the level of a representative element of volume. The three-dimensional, meso-scale thermomechanical model is applied, aiming to simulate the first stage of the coking process. Qualitative results for the yield of light gases and tar are reported and discussed. Ó 2017 Elsevier Ltd. All rights reserved.

Keywords: Granular media flow Heat transfer LBM Pyrolysis Chemical kinetics

1. Introduction Coking plants are widely used in coal processing industry to obtain chemically cleaner coal (coke), benzene, ammonium hydroxide and other chemical species (generally known as coke oven gas) as well as tar. Although the process has been used for many years (Serio et al., 1987), its detailed description is of recent research interest (Liu et al., 2013; Polesek-Karczewska et al., 2013; Maffei et al., 2013). Aside of fluid flow and heat transfer in a granular medium, chemical reactions (heterogeneous as well as homogeneous) have to be accounted for, including in particular the gas release from grains. In the meso-scale approach, these phenomena are modelled at the single pore level. Another issue, still to be dealt with, is the evolution of the interstitial space due to change of size and shape of coal grains. Miura and Silveston (1980) present an

⇑ Corresponding author. E-mail addresses: (J. Pozorski).

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(A.

http://dx.doi.org/10.1016/j.ces.2017.06.017 0009-2509/Ó 2017 Elsevier Ltd. All rights reserved.

Grucelski),

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analysis of coking with the emphasis on changes in the pore geometry during heating. Although the pyrolysis-like processes (of coal, biomass, waste matter, etc.) can be considered on different levels of description, still the most common way is to set up simulations at the macroscopic or averaged level. A number of papers review the numerical tools used for computations at this level. For example, Di Blasi (2008) deals with the physico-chemical modelling of wood and biomass. Averaged simulation results are presented in Guo and Tang (2005) for general process description in the coke oven. Otherwise, detailed studies of the coking process as well as the investigation of the coal bed geometry are reported in Keyser et al. (2006), including the effect of the coal particle size distribution on pressure drop. The averaged modelling has to be supported by the analyses at the level of a single grain that, in turn, result from a detailed description of the phenomena occurring at the micro scale level, i.e. accounting for the grain internal structure. Such an investigation is presented in Liu et al. (2013) and Wardach-S´wie˛cicka and Kardas´ (2013) where original models of heat and mass transfer are reported together with results of simulation at the single grain level. The

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A. Grucelski, J. Pozorski / Chemical Engineering Science 172 (2017) 503–512

processes occurring inside the grain are also modelled with different tools: Liu et al. (2013) use a direct numerical simulation along with multiple-reaction model supported by closure relationships for heat transfer and evolution of volatiles. The work by WardachS´wie˛cicka and Kardas´ (2013) was driven by similar motivation; the authors used a commercial software to describe the processes ongoing typically at the pore level, supplemented by closure relationships. A detailed model for coal particle oxidation was proposed by Beckmann et al. (2017). During the coking process, complex phenomena occur at the level of a single coal grain (Di Blasi, 2008). From the fluid thermomechanics point of view, the process takes place in a bed of granular material that, upon heating, ultimately becomes a homogeneous porous medium. Chemical reactions play a crucial role in accurate description of the pyrolysis and coal coking processes. Moreover, the release of coking gases into the volume of fluid has its influence on thermal dilatation of grains. Different models for chemical species evolution exist in the literature on coal devolatilisation; important conclusions concern a complex dependency of the pyrolysis process on the type of used coal, the heating rate, the maximum temperature, packing of the bed, etc. (Serio et al., 1987). The number of factors affecting the progress and the final products of pyrolysis makes it very difficult to provide a generalised description of the process. This is one of the reasons why the multilevel simulation techniques (for summary see Di Blasi, 2008) are increasingly often considered. Our longer-term objective is a physically-sound simulation of the coking process in the macroscale, applying one- or two-dimensional, unsteady formulation (Polesek-Karczewska et al., 2013). To address the problem, some closure relationships are needed; they can be obtained from numerical experiments in the mesoscale. It seems that the coupled subprocesses cannot be effectively simulated by more traditional computational fluid dynamics (CFD) methods, mostly because of the complex geometry and the need of coupling with phenomena occurring at the level of pores inside the grains. The Lattice Boltzmann method (LBM) is one of the alternative tools in CFD, applied among others for modelling the fluid thermomechanics in porous media. It is based on the Boltzmann equation with subsequent discretisation (Aidun and Clausen, 2010; Succi, 2001). The method has proven suitable for simulation of viscous and nearly incompressible fluid flows in simple and complex geometries (Succi, 2001; Chen and Doolen, 1998; Pan et al., 2006; Chai et al., 2010; Grucelski and Pozorski, 2013), as well as heat transfer (He et al., 1998; Wang et al., 2007; Grucelski and Pozorski, 2015), also with addition of chemical reactions (Di Rienzo et al., 2012). In the context of practical application, Asinari et al. (2007) reported the LBM results from modelling of complex phenomena in fuel cells at the single pore level for prediction of the overall system behaviour. The available literature demonstrates the reliability of the LBM computation results, so the method is perceived as a promising CFD tool for the multiphysics simulations in complex/random geometries. The main aim of the present work is the numerical modelling of fluid thermomechanics in a stationary granular medium with the chemical species release taken into account. The model is presented in Section 2. This process is considered at the level of a few grain lengths in a representative element of volume (REV). The extended LBM is applied to numerically solve the meso-scale model (Section 3) so as to simplify the complex phenomena in the micro-scale (like heat and mass transfer inside the coal particle and related chemical reactions). To handle this complex task, the modified method based on functional groups as chemical mechanism (Section 2.1) is applied. A validation case for the single grain is presented first (Section 4.1), followed by the results for chemical species evolution in the geometry of the 3D granular bed of spherical coal particles (Section 4.2).

2. Model of devolatilisation 2.1. Chemical mechanisms The release of gaseous species plays a crucial role in the coking process and the development of suitable numerical tools for devolatilisation modelling continues to be of recent interest. In the literature, at least four classes of models can be distinguished as used for a wide range of coal types. The first devolatilisation model, developed by Kobayashi (1976) and widely used in CFD studies (Authier et al., 2014), includes two competitive reactions in a function of temperature and heating rate. The second approach to chemical calculations is the FLASHCHAIN model (Vascellari et al., 2013); its clue is to represent coal as a composition of molecular chains, from very short up to virtually infinite ones. The initial setup is based on the atomic distribution in a fossil coal sample, applying a statistical technique to create the molecular chains. The model also introduces four chemical devolatilisation reactions; tar and gas yields are described in detail in the specification of FLASHCHAIN. As the third approach, the chemical percolation for coal devolatilization (CPD) model has recently been developed for a rapid rate devolatilisation of coal (applying high heating rates, Chen and He, 2011). The model is suitable for coking process and it can also be implemented for advanced modelling of coal combustion. The main advantage of the CPD formulation is a direct application of measured features of chemical structure of coal. Although the model does not use any fitting coefficients, the reported results can accurately account for the release of tar and gaseous products. The fourth proposal, mainly considered here, is based on the approach of functional groups (FG) as applied to Depolymerization, Vaporization and Cross-linking, known as the FG-DVC model. It is described in detail in the work of Solomon et al. (1988), together with further developments. The model was also generalised by Zhao et al. (1994). The proposed procedure of interpolation uses the FG-DVC method in the devolatilisation modelling of different coals. In a nutshell, the procedure is based on the calculation of FG amount from the Krevelen diagram with imposed interpolation grid (Zhao et al., 1994). From the chemical standpoint, the main idea of FG is to represent the fossil coal particle as a set of FGs, like carboxyl, hydroxyl, methyl, and methoxyl, see Fig. 1. Every FG, in the function of temperature, releases specified chemical compounds, mostly light gases like CO, CO2 , H2 , CH4 , etc. Then, some amount of particular species participates in the process of tar creation in a specific temperature range. The reaction rate constant is described by the Arrhenius equation with coefficients presented in Serio et al. (1987) and Solomon et al. (1988). Additionally, species are specified in terms of the bond strength in the group (as loose, tight, etc.), resulting in an additional differentiation of coefficients in the rate equation. An example of the coal input data with composition of functional groups is given in Table 1. The order of FGs in the table is related to the ease of species release. A particular FG contained in the solid grain (s) is represented by a number of moles of a k specie, nk js ¼ Y k0 js qs V s =M k , where Y k0 is the initial mole fraction of FG inside the grain, qs is the density of coal, V s is the volume of grain and M k is the molar mass of the specie. The release rate of k specie is:

dY k ¼ Y k kk0 exp ½Ek =ðRhhiÞ dt

ð1Þ

where the pre-exponential factor kk0 and the activation energy Ek are taken from Serio et al. (1987); hhi is the grain averaged temperature.

505


Fig. 1. Schematic plot of reactions occurring for two FGs: carboxyl and aliphatic. Gases are released as the effect of transport off the heated coal grain; the gaseous products are then involved in formation of tar. Subscripts el; l; t stand for extra loose, loose and tight, respectively, to distinguish the bond strength (not discussed here; for further details see Serio et al., 1987; Solomon et al., 1988). The tar output, quantified by coefficient a, depends on the functional group.

Table 1 Percentage amount of functional groups in a grain of coal (a variety classified as the Zap North Dakota, Serio et al., 1987; Chen and He, 2011). Sulphur is omitted in the model; hydrogen cyanide and ammonia are treated as a part of the aliphatic group. Bond strengthnFG [%]

Carboxyl

Hydroxyl

Ether

Methyl

Aliphatic

Aromatic H

Aromatic C

Extra loose Loose Tight Extra tight

6.5 3 0.5 –

– 6.1 3.3 –

– 6 4.4 9

– 1.6 0.9 –

– 0.6 1.2 9.5

– – – 1.7

– – – 44

Two assumptions should be pointed here. First, we use the average temperature of a particular coal grain (the averaging in LBM is done over every solid node, see Section 3.1). During heating, changes of volume and size of the grain are observed; these are connected with release of gases. Due to the directional heating of solid grains the release of gases and thermal dilatation depend on the temperature distribution within the grain. For a detailed description of the process, the gas release should take this distribution into account, resulting in a non-uniform flux of gases on the grain surface. Nevertheless, in an industrial coking plant, grains have small diameters (typically 1–4 mm); therefore the simplification to take the averaged temperature in Eq. (1) seems justified for simulation at this level. Second, we assume that the gases released inside the grain at a given time step are immediately transported to its surface. Both assumptions simplify the light gases release model: the resulting mass flux is thus uniform over the LBM nodes of the given grain surface. To justify these assumptions, let us consider the Biot number for mass transfer, Bi ¼ aL=Ds , where Ds is the diffusivity coefficient for gases (coking products) inside the solid grain, L is the characteristic length (here, the grain size) and a is the mass transfer coefficient in fluid. It is noted that in the limit of Bi ! 0 the description can be simplified to the lumped model where the mass transfer inside grains can be ignored. Indeed, the diffusivity coefficient Ds for gases is high and the characteristic length is small for considered particles. Somehow challenging could be the estimation of the mass transfer coefficient a. Gases released during the time step are then transported into the fluid (with suitable formulae for the mass flux at the grain surface, as detailed in Section 2.2). From the application point of view, we model the transport of the specie with change of the mass of released gases calculated from:

dmk ¼ Mk nk js dY k ;

ð2Þ

where the expression nk js dY k involves summation over each FG containing species k. Information about the elemental distribution

and the remaining functional groups in the solid is updated at each time step. For each element j 2 fC; H; O; S; Ng, we calculate:

dmj ¼ M j

X nj M j dY k ; Mk nk js k

ð3Þ

where the sum extends over every specie k containing element j; for carbon, j ¼ C, the summation would involve a group of species fCH4 ; CO2 ; COg. In the above equation, nj is the number of atoms of element j in specie k. Additionally, for carbon C we include fixed carbon. Changes of physical properties of the granular medium due to heating and evolution of the distribution of elements (when species are released) are considered for possible next-term implementation. In Grucelski (2016), a chemically simplified approach to the coking chamber was presented, as compared to the one described here. There, a much larger number of grains were modelled, with some of the physical properties being a function of temperature, like the coal density and thermal properties of the solid, taken from Authier et al. (2014). Due to the simplification regarding chemical species evolution, only a mixture of species released over the coal grain surface is described. In such case, the choice of either the detailed model (based on atomic composition) or the approximate model (based on a local temperature in solid, for example see Polesek-Karczewska et al., 2013; Grucelski, 2016) depends on application and remains an open issue. The computation of tar amount released during the devolatilisation remains a complex problem, mainly due to transport process through the porous structure of the grains. Such a physically sound simulation in the REV becomes far too expensive for granular media, since at least several dozens of particles in the main direction of heat transfer have to be considered. That is why a simplified model of tar generation is put forward here. In our proposal the sum of gases released at each time step is used to calculate the tar yield from the coal grain. The amount of light gases that are

506


transported onto the surface in a time step, denoted as dY k , is computed from Eq. (1). We propose to calculate the released tar as

dmT ¼

X c MT T dY k Mk k

ð4Þ

where the subscript T stands for tar and the reaction rate for tar creation is dY T ¼ dmT kT expðET =RhhiÞ. Similarly as in Eq. (1), the values of ET and kT are taken from Serio et al. (1987). The conversion of a particular specie (from the gas yield, see Serio et al., 1987) during transport to the grain surface has to be accounted for when the final yield is calculated. As stated in the literature, the tar conversion factor is limited to about cT ¼ 25% of the gas yield at time step. Next, tar is uniformly emitted from the grain surface, see Section 3.1. From the algorithmic point of view, once the calculations of element distribution have been performed at a given time step (before determining the amount of tar to be released), we assume that only the prescribed amount of light gases contribute to the generation of tar. Tar is next subject to homogeneous reaction, i.e. the breakage of tar molecules composed of higher hydrocarbons. This step of pyrolysis is described in detail by Boroson and Howard (1989). Although our idea is to model the main part of pyrolysis, omitting the metaplast phase and secondary pyrolysis, the tar cracking procedure is implemented to check the overall correctness of the calculations. Because the model proposed in Boroson and Howard (1989) is valid for biomass, some modification is needed for the purpose of the present work. Namely, the release of carbon mono- and dioxide is ignored during the tar breakage. This is justified by the fact that these compounds are tightly bounded in the coal structure and the release enhancement of carbon oxides (reported in Serio et al., 1987) is mainly caused by the burnout of the coal particle and should not be attributed to the tar molecules breakage. 2.2. Governing equations

@ t u þ u ru ¼ q1 rp þ r mru;

ð5Þ

where the macroscopic fields are density q ¼ qðr; tÞ, pressure p ¼ pðqÞ and velocity u ¼ uðr; tÞ; m stands for the kinematic viscosity. The temperature field, h ¼ hðr; tÞ, is solved from the energy equation in fluid and solid (with the respective thermal diffusivities as and af ):

@ t h þ u rh ¼ r aff;s rh þ Sh ðY; hÞ:

ð6Þ

The macroscopic evolution of the chemical species Y k ¼ Y k ðr; tÞ, k ¼ 1; . . . ; K, is solved from the advection-diffusion equation in a volume of fluid:

@ t Y k þ u $Y k ¼ r Dk rY k þ Sk ðY; hÞ

3.1. LBM formulation for thermomechanics of reactive fluid flow The Lattice Boltzmann equation, discretised in time, space, and in advection velocity ei ; i ¼ 0; 1; . . . ; Q 1, on a regular square lattice, describes the evolution of a relevant physical field in terms of its distribution function, see Aidun and Clausen (2010). For threedimensional (3D) computations reported in the paper, Q ¼ 15 stands for a number of discrete advection directions, resulting in the so-called D3Q15 discretisation scheme. The evolution of the distribution function over a time step dt is tracked with a generic equation: eq vi ðr þ ei ; t þ dtÞ vi ðr; tÞ ¼ s1 v ½vi ðr; tÞ vi ðr; tÞ þ Q v;i ;

ð8Þ

where vi is a respective distribution function (in advection direction ei ), sv is a corresponding non-dimensional relaxation time, Q v;i is the source term, and veq i represents the equilibrium state of the distribution function. The form of equilibrium function corresponds to the macroscopic fields: (i) the LBM variant for the flow density and velocity is solved in terms of the density distribution function f i , as described in Grucelski and Pozorski (2013); (ii) the temperature field is found from the additional internal energy density distribution function (IEDDF) g i , see Grucelski and Pozorski (2015); (iii) the evolution of each chemical specie k ¼ 1; . . . ; K is governed by a separate distribution function /i;k , see Aidun and Clausen, 2010. eq

As an efficient approach in the mesoscale, especially promising for the multiphysics simulations mentioned above, we have chosen the lattice Boltzmann method (LBM) described in Section 3.1. The fluid thermomechanics in the present case is governed by the Navier-Stokes equations:

@ t q þ r ðquÞ ¼ 0;

medium as well as heat transfer (see Grucelski and Pozorski, 2013, 2015). Some details regarding chemical species transport with an averaged (single specie) model of devolatilisation were also presented in Grucelski (2016); in that work we used variable material properties of solid and fluid in LBM modelling. The LBM implementation presented in the following is a computationally efficient way of solving the system of macroscopic equations, Eqs. (5)–(7), see Succi (2001) for a formal proof.

ð7Þ

where the source terms for species are computed according to submodels specified in Section 2.1; these terms are further detailed in the LBM setting (Section 3.1). 3. Lattice Boltzmann method and computation setup The present work is thought as a next step in the development of detailed description of the coking process in REV with use of LBM for modelling at the grain scale level. Our earlier works concerned details of utilizing the LBM for fluid flow through a granular

In the case of fluid flow vi ¼ f i and also veq i ¼ f i in Eq. (8). The equilibrium state is a function of q and u at ðr; tÞ and has the following form:

eq eq f i ðr; tÞ ¼ f i ðq; uÞ ¼ qXi Ai þ Bi ei u þ C i ðei uÞ2 Di ðuÞ2 ;

ð9Þ

where u ¼ uðr; tÞ is the local fluid velocity; the coefficients Ai through Di in general depend on the modelled phenomena, discretisation scheme and direction i (see He et al., 1998; Di Rienzo et al., 2012); the weight coefficients Xi are specific for a given LBM discretisation scheme but independent on the distribution function being solved for (f ; g or /k ), see Succi (2001), Chen and Doolen (1998). Coefficients in Eq. (9) for the case of fluid flow are constant for each direction, irrespective of the discretisation model: A ¼ 1; B ¼ 3; C ¼ 4:5; D ¼ 1:5 (subscripts i are thus omitted). In the case of fluid flow, the relaxation time is:

sm ¼ 0:5 þ

m c2s dt

:

ð10Þ

The macroscopic fields are obtained by integration of the distribution function; in the discrete setting:

q¼

X f i; i

u ¼ q1

X f i ei : i

As already mentioned, q and u satisfy Eq. (5). Equations for heat transfer in LBM are solved in a similar fashion and it is shown (see He et al., 1998) that they satisfy Eq. (6). For the purpose, the internal energy distribution function vi ¼ g i is introduced; it is governed by Eq. (8) with the respective relaxation time sa;m and the equilibrium distributions g eq i . For heat transfer


with use of IEDDF, the thermal relaxation time for phase m has the following form:

sa;m

1 km ¼ 0:5 þ 2 qm C p;m c2s dt

ð11Þ

written both for fluid f (gas) and solid s (coal grains) in the computational domain, so m 2 fs; fg, C p;m is the heat capacity of m and km the heat conductivity. In the modelling of technological process like devolatilisation, both density and heat capacity for solid (qs ; C p;s ) depend on the atomic composition of the grain of coal. In our work, these material properties are treated as constant. For heat transfer, the macroscopic field in Eq. (9) is the temperature h and the coefficients are (for D3Q15 discretisation scheme):

8 i ¼ 0; > < f0; 0; 0; 3=2g; fAi ; Bi ; C i ; Di g ¼ f1; 1; 9=2; 3=2g; i ¼ 1; . . . ; 6; > : f3; 7; 9=2; 3=2g; i ¼ 7; . . . ; 14:

ð12Þ

The temperature at each lattice node is obtained by averaging of P IEDDF, h ¼ g i . Heat transfer modelled with the use of IEDDF is i

governed by Eq. (6), for details see He et al. (1998). Full description of equations in 2D and 3D, with exact arrays of coefficients in the equilibrium distribution, Eq. (9) for fluid flow and heat transfer can be found in Wang et al. (2007) and references therein. It is worth to mention that Eq. (9) has the same polynomial form in the case of evolution equations for density, internal energy and chemical species. In the present work, as a compromise between efficiency and accuracy, we choose D3Q15 discretisation scheme for the advection directions of the distribution functions f ; g and /k . The LBM boundary schemes corresponding to the specified physical boundary conditions are briefly described in Section 3.3. The chemical species evolution in case of released gases is also modelled by LBM with additional distribution function /i;k for every chemical specie k considered; the governing equation is again Eq. (8) with a corresponding non-dimensional relaxation time:

sD;k ¼ 0:5 þ

Dk wc2s dt

ð13Þ

where Dk is the diffusivity coefficient of k specie and w ¼ q =q is the ratio of a minimum density of fluid in the entire domain and the density at a given lattice node (Di Rienzo et al., 2012). The gas release process at the surface of grains is described in Section 3.3. As the initial condition for chemical species we impose zero concentration in the volume of fluid. During heating of the grain (by a flow of the hot fluid) chemical compounds are released at its surface. We have observed considerable jumps in concentration values occurring there (at the fluid-solid interface). Because of numerical instabilities that originate in the regions of large gradients of resulting macroscopic fields, typical of Eulerian methods, Di Rienzo et al. (2012) proposed a new numerical scheme to account for variable density field in the volume of fluid. Those authors presented results for flow in a 2D channel with chemical reaction controlled by a single global equation, using a modified equilibrium distribution function for D2Q5 discretisation scheme and proposed a modified distribution function for D2Q9 scheme. We have confirmed those findings (not shown here). In the present work, the problem in 3D case is remedied, inspired by a 2D discretisation scheme proposed by Di Rienzo et al. (2012) and the derivation of equilibrium function (see Asinari, 2006). For D3Q15 discretisation scheme we propose:

/eq 0;k ¼

507

where /eq i;k is the distribution function at equilibrium corresponding to k specie in i direction. The source terms for temperature and species, see Eqs. (6) and P (7) respectively, have the discrete form Sh ðY; hÞ ¼ N 1 k

DHk dY k ðY k ; hÞ and Sk ðY; hÞ ¼ N 1 The quantities s dY k ðY k ; hÞ. dY k ðY k ; hÞ should be understood as the release rates of species governed by Eq. (1), depending themselves on the species mass fraction within the solid grain. The source terms are further translated into the LBM formalism as respective Q v;i in Eq. (8). As the energy is released uniformly within a solid grain, N stands for the number of lattice nodes representing the grain. On the other hand, the species release takes place on the grain surface, hence N s is the number of grain surface nodes. Moreover, DHk is the enthalpy change per mole of the product (light specie from cracking of the respective functional group, occurring at h) calculated for nodes of a particular grain. In the LBM implementation of the chemically reactive flow, the source term for IEDDF of the mixture P of released gases in Eq. (8) becomes Q h;i ¼ HXi k Sk where the source term Sk is the released concentration of compound k which is formed at the time step in a given grain and H is the averaged enthalpy of formation. For the sake of simplicity, the influence of the latent heat due to the breakage of tar molecules is ignored. 3.2. Computational domain The industrial process, being the ultimate goal of the present model developments, is characterised by a large span in length scales from the pore size in a single grain of coal (or biomass), through the grain radius, through the size of the representative element of volume (REV), up to the scale of coking plant. Here, we present and discuss the results concerning the meso-scale (the grain size level) modelling of coupled phenomena occurring during devolatilisation in REV. In such a domain, much smaller than the length scale typical of industrial coking plant but much larger than the grain size of fossil coal, we create a random granular medium, see Fig. 2. The REV consists of several grains (typically, 40–60). The grains are assumed to be spherical with the mean Sauter diameter of a few milimeters; for a more detailed study on shape of coal particles see (Keyser et al., 2006). The radii of the spheres are taken from the clipped normal distribution of the standard deviation equal to two nodes distance on the lattice. The locations of centres of spheres are then taken from the uniform distribution in the computational domain. The spherical grain created in this way is next placed in the domain; if the new grain overlaps with any of the existing ones, then a new position is randomly chosen. The process is iterated a few times with a predetermined stop criterion. For a granular medium created in this way, the resulting values

qY k

ð9 7wÞ; /eq i¼1;...;6;k 9 qY k qY k ¼ ðw þ 3ei uÞ; /eq ðw þ 3ei uÞ; i¼7;...;14;k ¼ 9 72

ð14Þ

Fig. 2. Exemplary geometry of a few solid particles; the grey level depends on the particle diameter.

508


of porosity are e 0:7. We note that typical values corresponding to the packed bed are e 0:45. In our 2D simulations, the porosity of e ¼ 0:45 was attained (Grucelski, 2016). Once the required porosity is obtained, the created array of grains is projected on the regular LBM lattice. For a chosen range of radii, the spheres are rendered on regular lattice with satisfactory accuracy. The mean value of ratio of volume of the sphere discretised on the LBM lattice to the analytical volume is hV LBM =VðrÞi ¼ 0:98 and the standard deviation is r ¼ 0:02. Also the mean ratio of the surface area (discretised by LBM to analytical relation) oscillates close to one: hALBM =AðrÞi ¼ 0:94 with r ¼ 0:1. In the used algorithm, the surface of a sphere is composed of squares (cube walls); at the end, stair-case representation of the sphere is the result of that discretisation, see Kajzer and Pozorski (2017) for some remarks regarding the influence of curvilinear interfaces representation in LBM. Surfaces parallel and perpendicular to the axes of the coordinate system affect the final surface area: with growing resolution the deviation between analytical and discretised area does not decrease as fast as it is for volume. The simulation domain is fully periodic, so grains whose surfaces intersect the boundaries are copied to the opposite side of the domain. 3.3. Boundary schemes Although the exact grain surface is curvilinear inreality, in LBM we use a simple bounce-back scheme for the solid-fluid boundary with the no-slip condition for velocity. As the boundary condition for IEDDF, we use a scheme for temperature at the interface proposed by He et al. (1998). Results presented in the literature show a generally better accuracy of boundary schemes for curvilinear interface with solid-fluid interface tracking like the OSIF scheme, also developed for IEDDF (Grucelski and Pozorski, 2015). Yet, in the authors’ opinion, the gain in accuracy is not worth the algorithmic effort required to implement the scheme for a complex and random 3D geometry, as the one presented here. For the density distribution function f i , we use a boundary condition proposed by Zhang and Kwok (2006). The pressure profile at the inlet and outlet is conserved with the use of the distribution functions leaving the computational domain at outlet and inlet (respectively), weighted by a shift coefficient to reach the desired pressure drop over the REV length. Here, we use the pressure drop b ¼ pin pout calculated with the well-known Darcy equation where permeability is obtained from the Kozeny-Carman correlation (Chai et al., 2010) for granular media. Similar approach (with the Ergun correlation used) was earlier presented and validated by Grucelski and Pozorski (2013); here no additional validation is presented. Briefly speaking, the unknown distributions for density functions (with directions pointing into the computational domain) are functions of known distributions leaving the domain and the initial density q0 (see Zhang and Kwok, 2006): inlet

f in

outlet

¼ f out

inlet

q0 þ b=c2s ; hqoutlet i outlet

outlet

f in

inlet

¼ f out

q0 b=c2s : hqinlet i

ð15Þ

where f in and f in stand for unknown distribution functions where ei vectors point into the computational domain at the inlet (i = in) and outlet (i = out), respectively. To model the complex phenomena in granular media of representative geometry, one of the possibilities is to use the periodic conditions for both fluid flow and heat transfer. For fluid flow simulation we assume a constant pressure drop along the mean flow direction to get a steady state. For heat transfer, a shift-periodic boundary condition is used with a corresponding temperature difference between inlet and outlet as recently proposed by Grucelski and Pozorski (2016). Using one of the features of the proposed scheme with the temperature difference set at the beginning, heat

transfer is modelled until the steady state for fluid flow and chemical species transport is achieved and the periodic profile of temperature at the REV boundaries are preserved. In the case of chemical species, we distinguish two types of reactions: homogeneous (tar cracking) and heterogeneous (occurring in the grain volume). Heterogeneous reactions require modification of the boundary scheme applied on the grain surface. For chemical species concentration, right before the bounce-back b.c. is applied, the distribution functions are updated by adding the source term Sk (k ¼ 1; . . . ; K) with a required weight, Q k;i ¼ Xi Sk . So, for a node close to the solid–fluid interface, the increment of the distribution function due to release of gaseous products is d/i;k ¼ Q k;i . In case of surface nodes the bounce-back rule is applied after the streaming step to distributions denoted by /0:

d/is ;k ¼ d/0is ;k þ d/0is ;k ;

d/is ;k ¼ 0;

ð16Þ

in the above expression, we use the lattice symmetry ei ¼ ei ; moreover, is is the index of the lattice velocity in the solid node direction. 4. Results for species evolution The results presented here are obtained with chemical kinetics taken from the original paper of Solomon et al. (1988). The input data are taken for the coal classified as the Zap North Dakota with composition of functional groups already presented in Table 1. 4.1. Single grain As the benchmark case, we perform simulation of devolatilisation of a single grain. The method presented in previous section is used to model heat transfer and fluid flow with evolution of the species released from the grain. In the first step, we set coefficient cT , see Eq. (4), that stands for a fraction of the chemical specie that will be turned into tar following its transport to the surface of the solid grain. We have tested values of cT ¼ 0:2; 0:25 and 0:3; by trial and error we have found that for cT ¼ 0:25 the results nicely converge with the reference data. The computational domain is discretised using Dx Dy Dz ¼ 80 160 80 lattice nodes, with a single spherical grain placed at ðx; y; zÞ ¼ ð0:5Dx ; 0:25Dy ; 0:5Dz Þ and radius equal to r ¼ 0:125Dy : (i.e. about r ¼ 4 mm). The chosen lattice is a compromise between accuracy and efficiency. As a grid independence test, we have compared the total mass of released gaseous products for a few lattice resolutions. Some differences are still noticed (not shown here), yet they may result from the algorithm of species release at the approximately spherical surface. In Fig. 3 we present results for chemical species evolution in the flow for a few major species. Along the present LBM outcome, also results from computations by Serio et al. (1987) and experiment by Chen and He (2011) are shown for comparison. For the presented chemical compounds some discrepancies are noticed. The comparison should be taken with caution, since the discrepancies may be connected with different setups in the calculations and experiment. In the present work (similarly as in Serio et al., 1987), we use a low heating rate to increase the temperature of the grain. For CO2 , the overestimation is rather small and is possibly connected with release of the tightly bounded carbon dioxide from the carboxyl group. Overestimation in case of CH4 can be caused by setting, see Eq. (3), more methane to release as light gas (rather than in contributing to tar creation as done in more detailed simulations with FG-DVC model). It is visible that if some fitting were associated in tar cracking model, more accurate overall mass of released CH4 could be obtained. Other species like H2 O and CO are much more accurately predicted with the present method. Since simulations are carried out basing on the dry coal particles,


509

Fig. 3. Species release from coal particle: comparison of LBM results with reference data. For the particular chemical compound, the dashed line is a release rate whereas the solid line represents the total mass M gas released from the solid grain (left axis, as a percentage of the coal grain mass Ms ). For the total of released gases in the fluid volume: lines with squares are results from computations by Serio et al. (1987) and asterisks are results from experiment by Chen and He (2011). In case of H2 O the observed scattering is due to different initial condition: in computation of Serio et al. (1987) some initial amount of H2 O was introduced.

the initial level of water vapour is zero in the LBM calculations whereas in the reference data, see Fig. 3, the initial level is not exactly zero (possibly because of small amount of H2 O present in experiment). In case of CO, a different level of released gas is achieved when compared with the reference data; the increase of release of carbon monoxide occurs close to h ¼ 900 C where tightly bounded species are released. In reference data, the burnout of the sample is reported. Yet, a good agreement is obtained in the case of CO at lower temperatures. Due to the model used for tar release, a shift is visible on the temperature axis, see Fig. 4. The used simplification implies that the release of tar and light gases from the grain volume occur at the same moment. It is interesting to note that the shift is also noticeable when the LBM results are compared with results of the rapid pyrolysis case with the CPD model reported in Chen and He (2011). It could be interesting to apply LBM for modelling pyrolysis with a higher heating rate, possibly with recently developed models like CPD. It is important to mention that the applied lattice resolution requires to simplify the gas transport through the porous solid grain rather than to resolve it. The limitations of the in-grain transport process are not accounted for, so the tar yield (dashed line in Fig. 4) is much higher than the one reported in

Fig. 4. The amount of released tar normalised by the solid particle mass: LBM with and without tar crack (solid and long-dashed line, respectively), functional group (dotted line with +) and results of the CPD model (dashed line with +).

510


the reference work (see Serio et al., 1987, dotted line with points in Fig. 4). Because of that, the tar cracking model is implemented (see Section 2.1): the yield of the light gases caused by tar crack occurs in the bulk of fluid (not only next to the grain surface), so it has a rather homogeneous character. One of the advantages of the FGDVC model is lost in the proposed approach: by eliminating the off-grain transport, the resulting amount of the released species does not depend on the heating rate (the amount of the gases and the grain mass loss are identical for rapid and slow pyrolysis). Yet, the development of a general-purpose model is out of scope of this work. Since the overall agreement is preserved for slow pyrolysis, then the modified formulation can be used to model the release of light gases and tar from the heated coal grain. Fig. 5 presents the overall mass loss from the grain, obtained from LBM computations and compared with reference data (Serio et al., 1987) based on the fully implemented the FG method, and the prediction of the first-order model, also reported in Serio et al. (1987) and CPD results from Chen and He (2011). The experimental data are for the low-heating-rate (LHR) pyrolysis. The loss of coal grain mass during the LHR pyrolysis rapidly increases for temperatures h > 900 C due to burning of the coal particle (along with a high yield of carbon monoxide). For comparison, also results where rapid pyrolysis is modelled by the CPD model are presented. The release of tightly bounded carbon monoxide is possible in presented approach; it also causes an additional loss of particle mass for h > 900 C. The devolatilisation process begins at the same temperature for LBM as for the reference data. Next, the release of gaseous products occurs during the heating of the grains, the mass loss is modelled (with reasonable accuracy) using the LBM-FG approach for meso-scale simulations. For temperatures above 550 C, the mass loss computed with LBM becomes too high compared with experimental data and for h > 580 C, the mass loss is higher compared to the first order model. For temperatures above 550 C, the mass release of the coking products is notable for higher order models causing the increase of the mass loss in contrast to the first order model (Fig. 5, fine-dotted line with +). When h > 750 C, the LBM computations show a growing release rate of

Fig. 5. Comparison of LBM results with reference data for the overall mass loss of the coal grain. Present results: ‘‘basic LBM” stands for a variant of the model where FG is implemented; ‘‘modified LBM” relates to calculations with additional tar cracking. Results of the CPD model are taken from Chen and He (2011). Data from Serio et al. (1987): first order model with k ¼ 4:28 1014 expð5470=RTÞ s1 (finedotted line with +) along with experimental results for the low-heating rate (LHR) pyrolysis; the ‘‘functional groups” line stands for complex FG computations and ‘‘theory line” represents reference data (Serio et al., 1987).

gaseous compounds. In reference models (dashed and dotted lines with + symbols) the release is stopped around h ¼ 900 C because of burnout of the sample, whereas in LBM calculations, breakage of the tight bounds between functional groups and corresponding chemical species causes growing rate of gas release. For the modified LBM scheme (Fig. 5), the overall mass loss is smaller; this is caused by introducing the tar cracking model. 4.2. Packed bed of grains As the second case considered in the paper, we model the thermomechanics for flow past several spherical obstacles representing a granular bed of coal. We aim to simulate first stage of coking. The subsequent stages of the process, i.e. mechanical interaction of swelling grains and the formation of the porous coke structure, are left for further study. We assume a fully periodic domain (being a simple REV) with the grains not crossing the inlet or outlet boundaries (due to simpler implementation of periodicity condition). In case of side boundaries, it is important for modelled phenomena to implement the periodicity, with grains possibly crossing the boundaries. The mean diameter of generated spherical grains is 10.2 lattice nodes and the Sauter mean diameter is 12.5 lattice nodes. The smallest obstacles correspond to discretised spheres with diameter of eight lattice nodes. Because of algorithm used to create the granular medium (appending new grains until a desired porosity is achieved), there are more smaller grains. Additionally, none of the grains are allowed to cross each other and no interaction forces between them are accounted for. Similarly as for the single grain validation case (Section 4.1), the shift-periodic condition (see Zhang and Kwok, 2006) is used for fluid flow with assumed pressure drop over the extent of REV. This type of condition is also applied for heat transfer Grucelski and Pozorski (2016): the temperature difference over the REV is set to dh ¼ 800 C with the reference temperature taken as href ¼ hðy ¼ L=2Þ. At the side boundaries the fully periodic boundary condition is applied. Although radiation phenomenon plays a role in heat transfer in the coal bed (cf. Polesek-Karczewska, 2017), it is not considered here. During the heating, each grain releases some amount of gases on its surface. The evolution of volatiles is next dealt with by

Fig. 6. The total amount (the top plot) and the rate of creation/annihilation (the bottom plot) of tar into fluid, normalised by overall volatiles (amount at the end of simulation); the tar yield is plotted by dashed line. The lines at lower plot present the release of tar for two particular obstacles whose centres are located near the inflow and outflow, respectively.


511

Fig. 7. The yield of gaseous products from devolatilisation: four panels for CH4 ; CO;CO2 and H2 O. The LBM simulation: the overall amount of product in the domain (solid line), the average release rate in the fluid domain (long dashed line), and the release rate from obstacle close to the inlet/outlet of the domain (short-dashed and dotted lines, respectively).

LBM. Modification to the chemical model of devolatilisation, presented in Section 2.1, is clearly visible in tar creation process, see Fig. 6. Aside of the total amount of tar in the domain (normalised by the final amount of volatiles), also the release rate is plotted, both in the whole domain and for two chosen obstacles (with centres close to inlet and outlet, respectively). Information about the yield from a particular obstacle is collected during the creation of tar as well as other products of devolatilisation, without account for tar breakage reactions. The two obstacles considered in Fig. 6 difffer in size, so some differences in the release rate of gaseous products are noticed. Tar release from a single grain along with the layout of the obstacles brings important information about peaks in time record of the global yield in the REV. Moreover, during the evolution of chemical species, we observe some fluctuations in the overall tar yield. Due to implementation of tar cracking reactions after release from the solid grain, the amount of tar fluctuates and some time averaging may be considered. In the calculation we use the same time step for LBM and chemical kinetics. The present domain is taken only for the benchmark purpose: a physically sound calculation in a REV of the coking plant would demand a much larger domain and the time step dt 105 —106 s Grucelski (2016). Because of that, no

internal time loop for chemical kinetics is used. The overall, normalised amount of tar in the domain (solid line in Fig. 6) should correspond with the yield of tar; locally occurring instabilities (dashed line) are insignificant for the integrated amount of tar. The tar yield is noisy when recorded over short time intervals (no averaging here). Also, it should be beneficial to consider a greater number of grains in the domain. Yet, the present case is already quite expensive for desktop computations in the 3D setting. When compared with results for other devolatilisation products (Fig. 7), one notices that tar release starts at the same moment as the yield of CH4 and CO and continues until strongly bounded hydrogen is released (not shown here). This mimics the introduced changes, ie. calculating the tar yield basing on the release of other gaseous products. Similar results but for light gaseous products are presented in Fig. 7. The total yield and exemplary yields from two chosen obstacles (the same as in Fig. 6) are in agreement: the gas release starts when the hot fluid flows past the first obstacles and finishes when the last obstacle is reached. In the case of methane and hydrogen, the model assumes correction of the yield by the cracking of tar. The tar cracking, governed by additional reactions introduces a small deviation of the resulting yield at 22 6 thUi=dsauter 6 30 and

512


a change of concentration that occurs even when devolatilisation is finished. In the case of other light gases, the change of their total amount is minor once the release from the grains has stopped. For results gathered for granular media a small negative yield is observed in the domain after devolatilisation is finished and also after tar cracking is finished (for CH4 and H2 ). The negative yield is possibly an artifact caused by a relatively coarse mesh. 5. Conclusions and perspective We have presented an approach for meso-scale modelling of devolatilisation process in two setups: (i) in a simple geometry of a single grain of coal, for which a comparison with reference data is made and (ii) in a simulated granular medium. To the best of the authors’ knowledge, the presented methodology is a first attempt in LBM to couple the approach for chemical species evolution of Di Rienzo et al. (2012) with three-dimensional meso-scale thermomechanical modelling of the coking process. Chemical kinetics has been solved using the functional group method with addition of tar cracking reactions. The present approach also accounts for simplified modelling of gas and tar release from the coal particle. Presented results are mainly concerned about the yield of devolatilisation products and the overall mass of chemical compounds in the domain. In the validation case of single coal grain, a satisfactory agreement was achieved for light gases (carbon dioxide, methane, steam) whereas for carbon monoxide some overprediction is possibly caused by exposing the particle to high temperature which leads to release of CO tightly bounded to coal molecules. Concerning tar release as well as the mass loss of coal grain, good agreement with reference data was obtained. A delay in tar yield in function of temperature is caused by used simplification: the main yield is triggered by the release process of light gases. For mass loss even more detailed data were presented here than those from the first order model (Serio et al., 1987). In the case of granular media simulation, qualitative results of the model were reported for the total yield in the domain as well as for the gas and tar release rates from two chosen coal grains (with centres located close to the domain inlet and outlet, respectively). In the paper of Grucelski (2016), a model widely applied in the averaged modelling was used for calculation of the thermal conductivity in detailed (i.e. the grain level) calculations of flow thermomechanics in reacting granular media. However, that model depends on the local porosity and averaged temperature which blurs the level of details available in the meso-scale modelling, as demonstrated by the present results. We believe that the LBM model developed in the present paper offers an improved description of devolatilisation of solid fuels in the REV geometry coupled with a fast and accurate model of fluid thermomechanics on the level of a single grain. A natural follow-up of this work is to include a finer, yet computationally affordable, spatial resolution and/or a larger REV. Other recent approaches for coal devolatilisation, such as the CPD model, may as well be worth exploration in the LBM setting. As a next step towards a physically-sound simulation of industrial scale bed of coal particles and the coking process, we intent to consider a REV containing more grains and of lower porosity, corresponding to the packed bed. Also, following a preliminary attempt (Grucelski and Pozorski, 2012), the account for thermal dilatation of grains is of interest to describe the resulting mechanical stresses in the granular medium. Acknowledgement The present work was initiated during an internship of A.G. at the Politecnico di Torino, Italy, in the framework of the HPCEuropa programme, under the guidance of Professor Pietro Asinari.

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