Gas–solid conversion in fluidised bed reactors

Gas–solid conversion in fluidised bed reactors A. G´ omez-Bareaa,∗, B. Lecknerb, D. Santanac, P. Olleroa aBioenergy Group, Chemical and Environmental Engineering Department, Escuela Superior de Ingenieros, University of Seville, Camino de los Descubrimientos s/n, 41092 Seville, Spain bDepartment of Energy and Environment, Chalmers University of Technology, S-412 96 G¨oteborg, Sweden cThermal and Fluid Engineering Department, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Legan´es Madrid, Spain Abstract A simplified model for gas–solid reactions in fluidised bed (FB) is proposed. Such models already exist for catalytic gas–solid reactions (CGSRs), providing general description of the system in terms of main governing parameters. Expansion of this approach to non-catalytic gas–solid reactions (NCGSRs) is difficult, because the solid reactant takes part in the reaction. Therefore, FB reactor models for NCGSR are usually devised only for specific cases, and a general analysis has not been presented up to date. The present model allows analysis of different types of NCGSR in a generalised way, handling catalytic reactions as a particular, simpler, case. It is shown that the reactor behaviour can be described by three governing dimensionless parameters. Two additional parameters, quantifying the importance of diffusion effects in single particles are also identified, and their impact on reactor behaviour is analysed. Possible simplifications are explored. Model limitations, that is, assumption of isothermal bed and particle and the occurrence of only one reaction, are discussed. Examples are outlined to show the applicability of the method. Keywords: Fluidised beds; Gas–solid reactions; Transport processes; Modelling; Non-catalytic reactions; Diffusion effects 1. Introduction Many important processes, in which non-catalytic gas–solid reactions are involved, take place in fluidised bed. Typical applications are found in metallurgical and thermochemical conversion processes. Examples of metallurgical importance comprise the reduction of metals oxides (iron, nickel, etc.), roasting of ores of heavy metals in sulphide form, such as copper, nickel, zinc and lead. Thermochemical examples are combustion, gasification and pyrolysis of coal and biomass, including sulphur capture (in-bed desulphurisation) using mineral rocks, such as dolomite or limestone. To this class of processes belong also thermal decomposition reactions, fluorination of uranium and plutonium compounds, some granulation processes, etc. The optimisation and scale-up of these processes benefit greatly from modelling of the system. Detailed description of physical and Abbreviations: BFB, bubbling fluidised bed; CGSR, catalytic gas solid reaction; FB, fluidised bed; FDE, free of diffusion effects; NCGSR, non-catalytic gas solid reaction; SIM, sharp interface model; UCM, uniform conversion model. ∗Corresponding author. Tel.: +34 95 4487223; fax: +34 95 4461775. E-mail address: [email protected] (A. G´ omez-Barea). chemical processes occurring inside an FB for NCGSR is, however, a difficult task. Simple methods providing approximate solutions for first estimates are quite useful. For instance, an approximate description of gas–solid reactions in FB can be sufficient for selection of mode of gas–solid contact, preliminary design, and optimal operating conditions by sensitivity analysis. Such models already exist for CGSR in FB [1–4]. In this type of system, the solids are unchanged as reaction proceeds and bed removal is not usually undertaken during steady-state operation if the catalyst is not poisoned. In contrast, simple models have not been developed for NCGSR due to their complexity compared to their catalytic counterparts. Although many FB reactor models have been published, they are devised for specific reactions only. A general framework for simplified treatment of NCGSR is not yet available. The performance of gas–solid reactions in FB has been described by several approaches. Early models treated the FB reactor as if the gas and solids were mixed, avoiding the multiphase nature of the bed. These ‘single-phase’ models assumed that the reactor performance was determined by the residence time of the gas. The breakthrough caused by the introduction of the two-phase theory proposed in the early 1950s, allowed . 1 Nomenclature A Non-reactive component in the solids (ash or equivalent) ATbed cross section (m2) Bi Biot number cgas concentration (mol m−3) C reactive component in the solids Deeffective diffusivity of the reactant solid particle (m2s−1) Dtbed diameter (m) DapDamk¨ ohler number at particle scale, defined in Eq. (30) DaRDamk¨ ohler number at reactor scale, defined in Eq. (22) DasDamk¨ ohler number for the solid reactant, defined in Eq. (40) ffunction f1(xc0,λ) function defined in Eq. (45) f2(xc0,λ) function defined in Eq. (48) F(xc) function expressing the dependence of dxc/dton xcfor any ηp Fi(xc) function expressing the dependence of dxc/dton xcwhen ηp= 1 (kinetic regime) F0,F1inlet and outlet flowrate of solids (kg s−1) gacceleration of gravity (m s−2) g(xc) function expressing the change of effective diffusivity with xc G gas reactant Hheight of the vessel containing the bed (m) knth-order kinetic coefficient in the kinetics ((−r)=kcn) ((kgmol m−3)1−ns−1) kbcoefficient of interchange between bubble and emulsion (s−1) kGexternal mass-transfer coefficient (m s−1) Krkinetic coefficient accounting for gas concentration and temperature (s−1) Lequ equivalent size of solid particle (m) Lfbed height (m) MThiele module, function of conversion Mcmolecular mass of solid reactant (kg kgmol−1) norder of reaction Naconcentration efficiency, defined in Eq. (23) NTU number of transfer units, defined in Eq. (24) pb(xc) distribution of conversion in the bed (mass basis) p0,p1distribution of conversion in the inlet and outlet streams (mass basis) Pbpressure drop across the bed (Pa) (−r) intrinsic reaction rate per unit of particle volume ((−r)=kcn) (kgmol m−3s−1) rc,b overall rate of reaction in the bed (kg s−1) (−R) observed reaction rate per unit of particle volume (kgmol m−3s−1) R(xc) reactivity of solid reactant (s−1) Repparticle Reynolds number sdummy variable of integration Sh Sherwood number ttime (s) Tbbed temperature (K) ugas velocity (m s−1) Vpparticle volume (m3) wbmass of A and C in the bed (wb=wA+wc) (kg) wcmass of solid reactant (C) in the bed (kg) wTb total mass of the bed (wTb =wb+winert) (kg) xcconversion of solid reactant in a particle xc,b average conversion of solids in the bed Xggas conversion Yc0 mass fraction of solid reactant in the feed Yc,b mass fraction of solid reactant in the bed zaxial coordinate Greek symbols αdimensionless parameter at reactor level, defined in Eq. (52) βdimensionless excess of flow, defined in Eq. (25) δkinetic parameter in Eq. (60), also Dirac’s delta function εporosity εbbubble fraction ((m3bubbles) (m−3bed)) ηeexternal effectiveness factor ηiinternal effectiveness factor ηpparticle effectiveness factor ηph interphase effectiveness factor κparameter defined in Eq. (59) λdimensionless parameter defined in Eq. (41) νstoichiometric factor of the reaction ξkinetic parameter (see Table 1) ρdensity of solid (kg m−3) τRsolid residence time (s) Θ(xc) function defined in Eq. (39) Subscripts b bubble, bed, average in the bed c reactive component in the particle crit critical e emulsion i intraparticle in inlet out outlet p particle r reaction s surface 0 initial, superficial consideration of the multiphase nature of the FB by means of a simplified description of two phases, in which the solids and the gas were distributed in the bed. Thereafter the concept of ‘contact time distribution’ was recognized as a key factor for taking into account the time of gas contact with the solid reactant [2]. May [1], Orcutt et al. [3] and Davidson and Harrison [4] used the two-phase theory of fluidisation for calculation of gas conversion in various isothermal FB catalytic gas reactors. 2 Table 1 Main models applied to NCGSR kinetics Name Abbreviation Fi(xc)Θ(X) Reference Volumetric model UCM 1 −xc−ln(1 −xc)[22] Grain model; or sharp interface model GM (SIM) (1 −xc)2/3 3(1 −(1 −xc)1/3)[23,24] Random pore model RPM (1 −xc)(1 −ξln(1 −xc)) (2/ξ)(1 −ξln(1 −xc))1/2 [25] Simons model SM (1 −xc)(xc+ξ(1 −xc))1/2 2arctgh((1 −ξ)xc+ξ)1/2 [26] Johnson model JM (1 −xc)2/3eξx2 cNAEF [27] Dutta model DM [1 ±100xξ1ξ2 cexp(−ξ2xc)](1 −xc) NAEF [20] Gardner model GM (1 −xc)e −ξxcNAEF [28] Chornet model CM √xc(1 −xc) 2arctgh(√xc)[29] Modified volumetric model MVM ξ1/2 1ξ2(1 −xc)[−ln(1 −xc)] NAEF [30] Traditional model TM (1 −xc)ξ(ξ−1)−1[(1 −X)1−ξ−1] [31] Polynomial model PM n  i=1 ξixc(1 −xc)iNAEF [32] The third column presents Fi(xc), the function modelling the behaviour defined in Eq. (1). The fourth column is the function defined in Eq. (39).ξiare kinetic model parameters; NAEF: not analytical expression found. Extensive reviews have been published on modelling of FB reactors [2,5–7], where the analyses were performed with different degree of sophistication. Several publications have surveyed the ability of FB reactor models in a variety of gas–solid reactions [5,8–11]. Generally, solutions of two-phase models based on isothermal catalytic systems have been presented in terms of two main dimensionless groups: one representing the dimensionless reaction rate and the other accounting for the interphase mass-transfer resistance [2,5–7]. Analytical solutions have been reported for simple kineticschemes, such asfirst-,second-,etc., order kinetic(seefor instance Table 11.5 in [2]). Solutions for more complex kinetics have been presented for catalytic reactions, based on conventional two-phase models and Kunii–Levenspiel’s model [12,13]. Expansion to include thermal effects has also been undertaken in catalytic systems for simple reactions [7], but this extension causes difficulties because of the complexity of the treatment even for the simplest reaction scheme. In FB catalytic systems the solids are unchanged as reaction proceeds (if no catalyst poisoning occurs) and the solids are only considered as a sink in the evaluation of the reaction rate. The reaction rate on the catalyst particle can be subjected to diffusion effects, but these do not change with time. For NCGSR in an FB, in contrast, the solid reactant is constantly consumed and solids make-up is required for steady-state operation. At any instant, the reactor contains particles that have spent different lengths of time inside the bed, and, thus, they have a wide burn-off distribution or particle age. During the course of reaction, the solid reactant contained in the particles is gradually affected, and the density and size of the particles change depending on the operating conditions in the bed. This behaviour can change from one particle-size fraction to another depending on the concentration of the solid reactant within the particles. A general description of the bed should account for variation in size and density of the reacting particles [14,15]. In addition to the aforementioned two main dimensionless groups appearing in the isothermal FB catalytic reactor, a third parameter taking into account the relative amounts of gas and solid reactants fed to the reactor is required to describe the NCGSR in an FB [9,10]. The abbreviation NCGSR represents heterogeneous reactions where the active solid participates in the reaction, in contrast to catalytic systems, which, if not poisoned, remain unchangedduring reaction. Somecatalyticeffectsmay existanyway caused by the inert material, for instance, minerals in coal or biomass particles in thermochemical processes. However, such effects are included in the gas–solid kinetics (expression dxc/dt, see Eq. (1)) determined in the laboratory, in this way being an input to the model presented. From this discussion it is clear that an FB reactor model for NCGSR should consider: (1) continuous bed removal; (2) variation of physical properties and reaction rate of single particles as reaction proceeds; (3) the distribution of conversion of the particles in the bed; (4) varying diffusion film and intraparticle mass-transfer limitations with burn-off. As a result, FB reactor models for catalytic reactions are not generally valid for NCGSR. The need for all these (and in some cases other) considerations is the reason why dedicated models have been developed for NCGSR in FB. Many reactor models exist, but they are devised solely for specific reactions. The reviews of Yates [16], Doraiswamy and Sharma [17], and Grace [2] survey the most popular models developed until the end of 1980s. In the present work, a method is developed for the solution of NCGSR in an isothermal FB, allowing analysis of general NCGSR by a common procedure. In this way, the simple modelling approach already existing for catalytic systems is extended to non-catalytic systems. The treatment considers isothermal conditions both in the phases and within the reacting particles, which imposes some limitations to the application of the method. Furthermore, some NCGSRs imply consideration of various heterogeneous reactions, and this could limit the method further. This and other limitations are dealt with at the end of this work where extension of the method and possibilities to overcome limitations are discussed. 2. Problem description and definitions Fig. 1 illustrates the problem dealt with. The gas reactant G is introduced into the FB reactor as part of the fluidisation agent 3 Fig. 1. (a) Model concept showing the hypothesis assumed in this work: the right-hand drawing of (a) presents the basis of the two-phase model developed showing the resistance to mass transfer between bubble and emulsion. Left-hand drawing of (a) is a zoom of the processes occurring in a reacting particle (with flat geometry for simplification), including the main resistances to mass transport: in the film (external mass resistance) and within the solid particle (reaction and intraparticle resistances). (b) Process scheme showing the contact pattern in an FB and population balance definitions. with a concentration cin. It passes through the bed as bubbles with a concentration cb, and through the well-mixed emulsion phase with a concentration ce. The G species is transferred from bubble to emulsion to reach the reacting sites within the reacting particles, where the reaction is C(s) + νG(g) →products. The resistances of transport and reaction and the main assumptions that have been made to develop the mathematical model are shown in Fig. 1a. The resistances are: bubble to emulsion resistance, external film resistance around the solid particle (the resistance within the emulsion phase is assumed to be concentrated around the particles), and intraparticle resistance. The inlet and outlet streams of the solids including reactant C are shown in Fig. 1b, where the bed inventory, wb, is also specified. The reactor contains particles that have spent different times inside the bed and, thus, have a wide distribution of conversion, pb(xc). This latter is considered equal to the distribution of the outflow stream, p1(xc), since perfect mixing of solid is assumed. We assume that a solid particle S, is made up of active solid reactant material C and non-reactive solid material A, ash or similar. In addition, there could be inert material, fed to the system for various reasons (for instance, sand to keep the bed constant). The conversion of the solid reactant C contained in S at any instant, xc, is defined as the relative difference between the initial amount of C and the instantaneous one, xc=(Yc0(t0)−Yc(t))/Yc0(t0). Ycis the mass fraction of solid reactant C in a given mass of material: Yc0 (kg C/kg S at xc0), and Yc(xc) (kg C/kg S at xc). Following these definitions (1 −Yc0xc) is the mass fraction of S in a stream of conversion xcand Yc0/(1 −Yc0xc)iskgCatxc0/kg S at xc. Note the difference between Ycand xc:Ycis an integral measure of the amount of C contained in a stream of material (or in the bed) containing various compounds (C+A+inert), whereas xcis a mark of the state of conversion of an individual particle refereed to their initial state of conversion, when it was fed to the reactor, xc0. Specification of xcfor a stream (or for the bed) has no meaning because, in the general case, in a given stream there will be particles with different degrees of conversion. However, the average of xcin a stream (or in the bed) is uniquely related with Yc(see Eq. (9)). The rate of conversion of a single particle due to chemical reaction, under chemical reaction control, may be expressed as [18,19]: dxc dt=Kr,eFi(xc) (1) Kr,e is the kinetic coefficient, accounting for the concentration of the gaseous reactant and temperature in the emulsion, where the reaction takes place. The function Fi(xc) expresses the dependence of the conversion rate on xc.Kr,e is evaluated for the conditions in the emulsion, where the reaction takes place. The rate of reaction can also be formulated as [20,21]: dxc dt=Mc ρc0 k(xc)cn e ν(2) kis the kinetic coefficient based on particle volume, and relates the rate of reaction per unit of volume with the gas reactant concentration, i.e. (−r)=kcn e. This definition is typical in GSCR where kis a constant for isothermal conditions. In contrast, for isothermal NCGSR kdepends on conversion. With the initial time as a reference, Fi(xc=xc0)=1 and k(xc)=k0Fi(xc), so for xc>xc0 one obtains from Eqs. (1) and (2): Kr,e=Mc ρc0 k0cn e ν(3) When diffusion plays a role the conversion rate of a particle is written as dxc dt=ηp(xc)Fi(xc)Kr,e(4) where the particle’s effectiveness factor ηp(xc) accounts for the diffusion resistance at particle scale (the external gas film and 4 the intraparticle resistance). Some authors have used expressions like Eq. (4) to estimate the role of internal diffusion [20,21]. ηp(xc) is defined as the ratio of the actual conversion rate of a particle to the rate free of diffusion effects (FDE): ηp(xc)=dxc/dt dxc/dt|FDE =dxc/dt Fi(xc)Kr,e (5) Formally, the Fi(xc) function should be free of diffusion limitations, i.e. it should be determined in the kinetically controlled regime. In this work Fi(xc) is the ratio of available solid surface at a certain conversion xcto that of a reference case, xc0.Table 1 provides some accepted models of Fi(xc) used for NCGSR reactions. From Eq. (4) a function can be defined: F(xc)=Fi(xc)ηp(xc), yielding dxc dt=Kr,eF(xc) (6) The overall mass rate of reaction rc,b in the entire bed is computed by rc,b=1 xc0 wbR(s)pb(s)ds(7) where wbis the mass of S (A + C) in the bed and pb(xc)isthe distribution of conversion in the bed (mass basis). The integrand wbR(xc)pb(xc) is the rate of reaction of solid particles in the bed having a conversion between xcand xc+dxc.R(xc)isthe reactivity expressed as kg C reacted/kg S at xcand time: R(xc)=Yc0 1−Yc0xc dxc dt=Yc0Kr,e 1−Yc0xc F(xc) (8) The solids can accumulate in the bed, depending on the net balance between the rates of feed, conversion, and removal of solids. A solid particle is fed into the reactor with an initial conversion xc0 and it is removed from the bed with a conversion, xc,b (average conversion of perfectly mixed particles in the bed). The fraction of C in the bed, Yc,b (kg C/kg S in the bed) is [9]: Yc,b=Yc0(1 −xc,b) 1−Yc0xc,b (9) Solving for xc,b gives xc,b=Yc0 −Yc,b Yc0(1 −Yc,b)(10) The bed material consists of the solid reactant C and the material remaining after completing the reaction A, ash or any other type of inert component originating from the feed stream. The mass of solid reactant C in the bed is the product of wband Yc,b obtained from Eq. (9): wc=wbYc,b(11) If inert material is fed to the bed, or if a batch of such material is used to fill the bed initially (for instance sand as initial buffer in biomass thermochemical conversion processes), an additional mass balance for this material has to be formulated. The composition of the bed at a given time depends on the way of operation. A detailed case by case analysis is out of the scope of the present treatment. In this work, at any instant, there are three amounts of materials in the bed: wc,wA, and winert.wbis the sum of wc and wA(wb=wc+wA), whereas wTb includes also the inert (wTb =wc+wA+winert), which is known by, for instance, pressure measurements (wTb =Pb(AT/g)). winert, if it exists, has to be calculated from an additional mass balance. The way to include this additional balance in parallel to the main problem is outlined in Section 4.3. 3. Development of the model 3.1. Modelling approach A realistic representation of the bed should account for the variation in size and density of the reacting material, as in the treatments by Chen and Saxena [14] and Overturf [15]. The model proposed aims at simplifying this general treatment. The method is based on two main steps: Step 1: Application of a fluid-dynamic model to a catalytic system. This model is derived without considering the non-catalytic nature of the reaction and the actual burn-off distribution in the bed. Step 2: Allowance is made for the deviation from the catalytic case, considering the extent of conversion in the FB by a solids population balance, which is solved by a kinetic model for a single particle. 3.2. FB reactor modelling for CGSR With the assumptions discussed in Fig. 1a molar balances for the gas in the bubble and emulsion phases lead to βu0dcb=kbεb(ce−cb)dz(12) (1 −β)u0(cin −ce)=Lf 0 kbεb(ce−cb)dz+νrc,b McAT (13) The boundary conditions are cb(z=0) =cb,in =cin (14) cout =βcb(z=Lf)+(1 −β)ce(15) The gas conversion Xgand the interphase effectiveness factor ηph are defined by ηph =ce cin n and Xg=1−cout cin ,(16) Integrating Eqs. (12) and (13), taking Eq. (16) into account, gives [33]: (1 −Xg/Na)n Xg/Na=Na DaR (17) where the parameters DaRand Naare defined in Eqs. (22) and (23). Combining Eqs. (16) and (17): Xg=(1 −η1/n ph )Na(18) 5 DaR=Xg ηph (19) These expressions can be combined to give DaR Na=Xg/Na (1 −Xg/Na)n(20) (1 −η1/n ph ) ηph =DaR Na (21) Eqs. (18)–(21) provide two independent relationships for four quantities: DaR,Na,Xgand ηph.DaRis the Damk¨ ohler number at reactor scale, expressing the relative importance of gas residence time and reaction time: DaR=ηpDaR,in with DaR,in =kcn−1 in u0/Lf (22) DaRis known for a CGSR if diffusion effects are absent, i.e. ηp= 1 (kinetic regime) because DaR,in is known. Here, we assume that the bed height Lfis known or can be determined by the pressure drop across the bed. In contrast, DaR,in is not known for a NCGSR even when the particles are in the kinetic regime, because the concentration of the reacting particles in the bed is unknown. We shall deal with this matter below. Nais the concentration efficiency in the one-dimensional bed, defined by Na=cin −cout cin −ce=1−βexp −NTU β(23) NTU is the number of transfer units and βis the dimensionless excess gas flow: NTU =kbεb u0/Lf (24) β=u0−umf u0 (25) The expression for βassumes that all gas in excess of minimum fluidisation velocity flows through the bed in the form of bubbles. This rests on the “two-phase theory” of fluidisation [12]. There is evidence, however, that there is a short-cut flow through the bubbles, especially in larger particle systems. This has been quantified in several models and correlations and depends on the groups of ub/umf,u0/umf and εb. Analyses of the throughflow in various two-phase models have been reviewed in [4,5,12,34]. The impact of throughflow on the prediction of gas conversion in simple and dynamic two-phase flow models was assessed by Mostoufi et al. [11]. A correction for throughflow could be needed when using βin Eq. (23), especially for larger particle systems. Sensitivity studies employing the final reactor model are helpful in identifying the need for further refinement. In the cases outlined in this work, the results have been found insensitive to this parameter. We conclude that the interphase effectiveness factor ηph is only a function of the group Na/DaR, and Xgis a function of the two groups Naand Na/DaR. The relationships needed to calculate NTU and βand related parameters depend on the flow pattern and the particle system under consideration. Examples of formulae useful for bubbling fluidisation in lab-scale FB can be found in Table 1 [33]. Further information for other fluidisation systems and scales is found in [12,34]. For reaction orders of 1, 1/2 and 2 explicit solutions for ηph are found in the literature [2,35]. In a general case, for nthorder kinetics, explicit solutions for ηph as a function of DaR and Na, can be obtained by Frank-Kamenetskii’s approximation (see Appendix A) applied to Eq. (21): ηph =⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ([{(1 −n)(DaR/Na)}1/n+1]n+n(DaR/Na))−1 with 0 <n≤1 2n[(2n)1/n−1+(1 +2n(DaR/Na))1/n]−n with 1 <n≤2.7 (26) Once ηph is known, the gas conversion is determined by Eq. (18) or (19). 3.3. Modelling of particle kinetics in CGSR To calculate the particle effectiveness factor ηpin the catalytic case, we assume that ηpdoes not depend on xcor, more useful for the later expansion to NCGSR, we consider that xc=xc0. The estimation of ηpis achieved through an internal and an external effectiveness factor, ηiand ηe, so that ηp=ηiηe. The detailed derivation of the equations needed and the way to obtain the necessary information from experiments have been published in [33]. The effectiveness factors for isothermal CGSR are written as ηe=cs cen (27) ηi=(−R) kcn s=1 Vpkcn sVp kcndVp(28) Under pseudo-steady-state conditions, the isothermal masstransfer problem for nth-order kinetics can be expressed as ηe=(1 −Dapeηp)n=(1 −Dapeηiηe)n(29) where a second Damk¨ ohler number, Dape, represents the ratio of the maximum diffusion rate (when cs=ce) to the reaction rate controlled by external diffusion (when cs∼0): Dape =kLequcn e kGce (30) Taking into account Eqs. (A.1), Eq. (29) can be approximately solved for ηe: ηe=[({(1 −n)Dapeηi}1/n+1)−n+nDapeηi]−1,0<n<1 2n[(2n)1/n−1+(1 +2nDapeηi)1/n+1]−n,1<n<2.7 (31) By solving the reaction-diffusion problem for a reactant particle, an approximate solution for ηiis often used: ηi=tanh(Ms) Ms (32) 6 Msbeing a conversion-dependent Thiele module: Ms=Lequn+1 2 kcn−1 s De1/2 (33) It is well know that Eq. (32) is strictly valid for first-order kinetics in a slab, but nevertheless, the generalised Thiele module defined in Eq. (33) makes the use of Eq. (32) a reasonable approximation for any geometry (characterised by Lequ) and reaction order, n[20,24,31,33,34]. In Eqs. (31) and (33) Msand Dape are evaluated for surface (index ‘s’) and emulsion (index ‘e’) conditions, respectively. The gas concentrations in the emulsion and at the surface differ from the inlet concentration, so they are not a priori known. Therefore Msand Dape should be related to known quantities, that is, they should be expressed as functions of quantities evaluated for gas inlet conditions (index ‘in’). This is done by taking into account Eqs. (16) and (28), yielding Dape =Dapin(ηph)(n−1)/n(34) Ms=Min(ηphηe)(n−1)/2n(35) Here ηph is calculated by Eq. (26), and ηeand ηiby Eqs. (31) and (32) where Msand Dape are calculated by Eqs. (34) and (35). There is a loop for ηph and the scheme of solution is iterative. From this treatment it is clear that the dependence of ηpis in the form: ηp=f(ηph,M in,Da pin,n) (36) The algorithm of solution and the graphical solution of this system is presented below in Section 4. The extension to account for the effect of solid conversion state is developed in Section 3.5. 3.4. FB reactor modelling for NCGSR In the model developed above for CGSR DaR,in was assumed to be known, given by the batch of catalyst in the bed and the properties of the catalyst. In NCGSR, on the other hand, DaR is unknown, because neither the amount of solid reactant nor its distribution of conversion pb(xc) in the bed are known. Consequently, a solids population balance should be formulated and solved. The definition of DaRfor NCGSR is better given in the following terms: DaR=∀xc∈VbwbR(xc)pb(xc)dxc u0ATcinMc/ν =rc,b u0ATcinMc/ν (37) Fig. 1b shows the main aspects and the nomenclature used. The main conditions were already discussed in Section 3.2.An additional assumption is that all the fines are returned to the reactor: there is no carryover, and all particles leave with the exit ash discharge. Also, all particles are assumed to enter with the same conversion xc0. The treatment follows the procedure developed by [36]. The present approach, however, expresses the equations in extent of conversion instead of time or particle size [37,10]. Following the nomenclature of Fig. 1a, a population balance over the reactor yields the distribution of the conversion of the solids (see Appendix B): pb(xc)=1 Das 1 F(xc) 1−Yc0xc 1−Yc0xc0 exp −Θ(xc) λ(38) Θ(xc) being a function Θ(xc)=xc xc0 ds F(s)(39) Expressions for Θ(xc), associated with well-known kinetic models are included in Table 1. The two dimensionless parameters Dasand λin Eq. (38) are defined as Das=Kr,ewb F0=Kr,eτR(40) λ=Kr,ewb F1=Kr,ewb F0−rc,b (41) Dasis the Damk¨ ohler number of the solid reactant, expressing the ratio of residence time of solids τR=wb/F0and reaction time 1/Kr,e. Note that Dasranges from 0 to 1 (the Das= 1 case is when the particles are made up of C entirely remaining in the bed just the time they need to react completely). The relation between Dasand λis obtained through the normalisation equation [36]: 1 xc0 pb(s)ds=1 (42) Eq. (42) is not satisfied, however, with the distribution calculated by Eq. (38) because pb(xc) should include both particles having solid reactant left (C + A) and particles completely reacted that still remain in the bed (only consisting of A). In Eq. (42) this second class of solids is not accounted for. Caram and Amundson [9,10] showed that for an FB coal gasifier (C ≡carbon + A ≡ash) an ash balance could solve this apparent difficulty (equivalent to and replacing Eq. (42)). The following treatment uses the same approach as that in [10]. An ash (A) balance over the system yields 1−Yc0xc0 1−Yc0xc,b=F0 F1=Das λ−1 (43) The C-concentration in the bed can be obtained by taking into account all particles having a C-concentration in the bed, Yc(s) given by Eq. (9). Integration over the bed using the distribution in Eq. (38) gives Yc,b=1 xc0 Yc(s)pb(s)ds=f1(xc0,λ) 1/Yc0 −xc0 1 Das (44) where f1(xc0,λ)=1 xc0 1−s F(s)exp −Θ(s) λds(45) Elimination of xc,b between Eqs. (43) and (44) yields Das λ=f1(xc0,λ)/λ+(1/Yc0 −1) 1/Yc0 −xc0 (46) 7 which is the equivalent to Eq. (42) and replaces that equation by accounting for the particles that have reached complete conversion and are still in the bed. An alternative equation (equivalent to Eq. (46)) is found by combination of Eqs. (38) and (7) and taking into account Eqs. (8),(39) and (41) to give Das λ=1 λ−f2(xc0,λ)Yc0 1−Yc0xc0 (47) where f2(xc0,λ)=1 xc0 exp −Θ(s) λds(48) Once Das,xc0 and Yc0 are given, Eq. (46) or (47) provide one equation for λ(or Das/λ). The distribution pb(xc) can be then calculated by Eq. (38), and the average conversion in the bed, Eq. (10), is computed by xc,b=1−f1(xc0,λ) λ(49) From Eqs. (40) and (41) rc,b becomes rc,b=1−Das λF0(50) The overall mass balance on the solids and gas reactant and the stoichiometry of the reaction link the conversions of solids and gas. By equalling the rate of disappearance of solids, rc,b/Mcwith the rate of consumption of the gaseous reactant, (cin −cout)u0AT/νone obtains Xg=1 α1−Das λ(51) where αis a dimensionless parameter defined by the stoichiometric ratio of the feed rates of the reactant gas and the solids: α=u0ATcinMc νF0 (52) Elimination of f1(xc0,λ) by combination of Eqs. (46) and (49) enables to relate xc,b and Das/λ: xc,b=xc0 +1 Yc0 −xc01−Das λ(53) Eq. (53) establishes clearly the boundary limits of Das/λwhich are obtained for the limiting cases xc,b = 0 and 1: Das λ∈1−Yc0 1−Yc0xc0 ,1 1−Yc0xc0 (54) Das/λnear zero means complete conversion of solids, whereas Das/λclose to unity stands for the case of null solid conversion. In the particular case when all particles enter the bed with xc0 =0 and Yc0 =1, Das/λis equal to one minus the solid conversion that is attained in the bed, that is 1 −xc,b. Thus, with xc0 =0 and Yc0 =1,Das/λranges from 0 to 1. Elimination of Das/λby combination of Eqs. (51) and (53) gives a relation between Xg and xc,b: Xg=xc,b−xc0 α(1/Yc0 −xc0)(55) The parameter Dasshould be evaluated for emulsion conditions, i.e. Das=Das,e, but the ratio Das/λdoes not depend on the reference situation for which the gas conversion is evaluated (see Eqs. (40) and (41)). The known parameter is actually Das,in (evaluated for the inlet conditions), but according to Eq. (16) Das(=Das,e) can be derived from Das,in: Das=Das,inηph (56) Hence, ηph has to be known to calculate Dasfor emulsion conditions. An expression for ηph results from Eqs. (18) and (51): ηph =1−1 Naα1−Das λn (57) Eqs. (51) and (53) allow the calculation of Xg=f(Das/λ,α, ηph) and xc,b =f(Das/λ). Taking into account Eq. (57) gives Xg=f(Das/λ,α,Na) so the reactor behaviour is governed by three parameters: Das/λ,α, and Na.Nacomes from the fluid dynamics (Eq. (23)), whereas αis obtained from available inputs (see Eq. (52)). The inventory of the bed, wbis known, for instance, from measurements of pressure drop across the bed (see Section 4.3). The group Das/λis obtained from Eq. (46) or (47). To apply these equations, the functions f1(or f2) defined in Eqs. (45) and (48) require the value of Θ(xc), defined in Eq. (39). Therefore F(xc) has to be integrated for all the degrees of conversion in the FB reactor. To undertake this estimation, a kinetic model should be established in order to have available the expressions Fi(xc) and ηp(xc). To sum up: for the estimation of Das/λby Eq. (46) or (47), a kinetic model should be formulated first and then solved for the conditions in the reactor. 3.5. Modelling of particle kinetics in NCGSR For the non-catalytic case the reaction rate of a particle developed in Section 3.3 has to be expanded to include the effect of conversion. This leads to the solution of a time-dependent problem with a moving interface within a particle. The rate of shrinkage/expansion of a particle’s external surface is difficult to generalise because it depends on the nature of the NCGSR. For instance, for gasification reactions, a threshold for the local conversion has been fixed at the instant when the ash layer of particle peels off [38,39]. This threshold condition allows theoretical computation of the particle’s boundary at any time. The threshold depends on type of reactor, resistance of ash, and operating conditions. In a FB, for example, the removal of an ash layer may be caused by attrition. In contrast, for reactions where a solid product is formed, the relation between the molar volumes of reactant and product is usually employed, together with some empirical parameter, to determine the rate of change in volume (shrinkage or expansion) [17]. In general, empirical input is needed at some level. Exceptions to this are the well-known uniform conversion model (UCM) and the sharp interface model (SIM) as we shall see later on, representing limiting cases. In sit- 8 uations where these extreme cases are not valid, general, but still simple, models could be applied. In the following such a model will be formulated using Eq. (4) together with an estimate of the effectiveness factor of the particle ηp(xc). As in the catalytic case, ηp(xc) is composed of an internal and an external effectiveness factor, ηi(xc) and ηe(xc). These are defined for NCGSR in the following way: ηe=cs cen =Kr,s Kr,e ,η i=dxc/dt Fi(xc0)Kr,s (58) As seen they depend on conversion. Eqs. (32) and (33) still apply. However, Msand Dape given by Eqs. (34) and (35) have to consider the effect of conversion, because k,Lequ,kGand De depend on it. The change with conversion can be followed by the variation in reaction rate, Eq. (4), and the change in diffusivity of the particle by g(xc), a function of the local porosity ε, and De0, the initial effective diffusivity. An empirical equation for g(xc) is usually accepted for gas–solid reacting systems [33,40,41]: g(xc)=De(xc) De0 =ε(xc) ε0κ =1+1−ε0 ε0xcκ (59) To estimate the change of particle size with xc, an additional relation is required. The shrinkage of a particle during consumption is not included in Fi(xc) that only measures the change of the internal surface. If a detailed model (integration inside the particle) is used, it is possible to establish the rate of shrinkage as shown by Srinivas and Amundson [38] and Morell et al. [39] for the case of gasification of coal particles. However, here we have formulated the model in terms of the global particle conversion xc, and the change in size with particle conversion cannot be calculated. Nevertheless, this information can be provided by a simple empirical equation: Lequ(xc)=Lequ,0(1 −xc)δ(60) where a judicious choice of the parameter δgives Lequ for any xc. Only in the limiting kinetic models, such as UCM and SIM, the assumption of an arbitrary value for δis not necessary: UCM implies a constant particle size (δ= 0) and in SIM, δ= 1/3. In between these two limiting situations, the more general progressive conversion model with changes in size and density can be applied by properly choosing a value of δin the range of 0–1/3. Now, with the definitions given in Eqs. (30) and (33), and with Eqs. (4),(59) and (60), the initial value Dapin,0 can be related to Dapin(xc) and Min,0 to Min(xc)as Dapin(xc)=Dapin,0[Fi(xc)(1 −xc)(3/2)δ] (61) Min(xc)=Min,0Fi(xc) g(xc)1/2 (1 −xc)δ(62) where a correlation for the external diffusion coefficient kGof the type of Sh ∝Re1/2 pwas used to derive Eq. (61) [43]. By Eqs. (31),(33),(61) and (62) the desired relationships are determined: Dape(xc)=Dapin,0[(ηph)(n−1)/n(Fi(xc)(1 −xc)(3/2)δ)] (63) Ms(xc)=Min,0(ηphηe)(n−1)/2nFi(xc) g(xc)1/2 (1 −xc)δ (64) In conclusion, Eqs. (31) and (33) allow calculation of ηe and ηiand so ηp. The quantities Msand Dape, appearing in these equations, are calculated by Eqs. (63) and (64). From this treatment it is clear that the dependence of ηpis in the form ηp=f(ηph,M in,0,Da pin,0,F i(xc),g(xc),δ) (65) being the NCGSR version of Eq. (36). 4. Discussion 4.1. Solution for CGSR The explicit solution for ηph found in Eq. (26), i.e. ηph =f(DaR/Na,ηp,n), is displayed in Fig. 2 by solid lines. It can be demonstrated that this solution includes as particular cases published analytical expressions, such as reported by [2] for the modified Orcutt model of irreversible reactions with n= 1/2, 1 and 2 (symbols in Fig. 2). The difference between the solid lines and the symbols is very small and entirely associated with Frank-Kamenetskii’s approximation used to derive Eq. (26) from Eq. (21). In fact, Eq. (20) is equivalent to the solutions of Orcutt’s model, but the present formulation provides an additional scheme for simple estimation of diffusion limitations at the particle scale and for expanding this scheme to NCGSR. The simplest case (n= 1) allows a straightforward physical interpretation of the solution. In this case the Fig. 2. Solution for gas–solid catalytic reactions in an FB drawn for the interphasic effectiveness factor, ηph as a function of DaR,inηp/Nafor various reaction orders, n(between 0.25 and 2), according to Eq. (26). (Solid lines represent the present model, whereas symbol lines are results obtained from Orcutt’s model.) The particle effectiveness factor ηpis unknown, so this figure must be used in parallel with Fig. 3 to determine iteratively ηpand ηph. 9 tion is often made. If thermal effects should be considered or not depends largely on the type of reaction, but also, on the operation conditions, the latter making it very difficult to establish general guidelines for the applicability of the model. Assessment of the presence of these potential thermal gradients prior to application of the model is, therefore, recommended. This can be made by estimation of the thermal Biot number and the maximum thermal gradient between the emulsion and particle from a heat balance over a reacting particle. Non-isothermal analysis at a particle scale, in general, requires two further parameters to account for the thermal sensitivity of the chemical reactions (the Arrhenius parameter) and to quantify the thermal effects relative to the heat conduction (the Prater number). If, in addition, the difference between the phases is important, further parameters have to be considered by formulating an energy balance over the reactor. To account for all these phenomena in a generalised formulation such as the one presented, complicates the presentation and makes less meaningful a comprehensive analysis of governing parameters. When more than one reaction occurs, or when the gas reactant and the product gas species further combine homogeneously or/and react with other compounds in the gas mixture, some extensions have to be made. An example is given to illustrate how reasonable simplifications can lead to application of the method. Let us consider, for instance, simultaneous CO2 and H2O gasification of char generated after devolatilisation of biomass. The relative amount of H2O and CO2depends on previous drying, devolatilisation, and combustion processes. These processes occur at much higher rate than the gasification of the char, and so they can be calculated uncoupled to char gasification. Char reduction processes can be simplified by considering that H2O and CO2and H2and CO are lumped into the same pseudo-components, R and P, respectively. As a result, the only heterogeneous reaction to be considered is char + R →P. This reaction is assumed to occur in the emulsion phase where most char particles are found. This scheme is justified by the similar stoichiometry of char-CO2and char-H2O and by the commonly assumed equilibrium of the WGSR (water–gas shift reaction). The relative concentrations of H2O and CO2are adjusted to the local thermal environment around the particles, because the WGSR is more rapid. The method of this work has been applied to such a case [44], giving close agreement with results from advanced models, shortening considerable the computations, and most importantly, reducing the input data needed for the calculations. In the method wb(=wc+wA) has been considered known. However, when there is inert material in the bed, winert, this can be calculated because a pressure measurement provides the total amount of bed, wTb and hence, also the amount of inert material (wTb =wb+winert). winert, can be calculated once the manner of operation of the FB system is specified. As an example, let us consider an FB operating with a constant solids inventory: a continuous drainage of bed material from the system is made, and so, to keep the bed material constant, a continuous makeup of inert material is needed. At steady state, a simple mass balance over the inert material yields an additional equation for winert. In general, the composition of the bed has to be known for the calculation of winert, so this equation is coupled to the main problem, because to solve the new equation, the composition of the bed has to be known, i.e. Yc,b. Therefore, Yc,b is assumed, winert is calculated, and then, also wb. If the new Yc,b differs from the assumed value, a new winert is estimated and the calculation is repeated until convergence. To sum up, the consideration of inert material introduces a second loop, which has to be solved in parallel with the main problem. In practise, two iterations are often enough to attain the solution because the main problem is not sensitive to this loop. 5. Summary and conclusions A methodology is proposed for evaluation of general gas–solid reactions in isothermal FB. A model is developed in two stages. First, a method for evaluation of gas conversion is formulated by applying the two-phase theory of fluidisation on FB catalytic reactors, in which only gas conversion is considered. A condensed formulation is given to calculate gas conversion as a function of the governing parameters. In a second stage, the model is extended to account for non-catalytic reactions by incorporating variation of particle properties and reaction rate with conversion, as well as the distribution of the conversion of reacting particles in the bed. Three groups govern the solid and gas conversion in the reactor: (1) the ratio of reactant gas and solid feed flowrates, α; (2) the concentration efficiency in the entire bed, Na; and (3) Das/λ, being an indication of the solid conversion. The group Das/λis obtained from a population balance taking into account the overall contribution of all reacting particles in the bed. A simplified kinetic model for a single particle is developed to characterise the governing parameters at the particle scale. Besides the intrinsic kinetics, two parameters are identified, quantifying the diffusion effects at the particle scale: a generalised Thiele module Min,0, and a Damk¨ ohler number at a particle scale Dapin,0, both taking zero conversion and gas inlet conversion as reference states making these parameters known quantities. Simplification is possible for limiting values of the three principal reactor parameters (α,Na, and Das/λ). The simplest case for NCGSR neglects the role of the distribution of conversion in the bed and allows obtaining a rapid solution for anykinetics.Examples confirmed thegood results of the method. Moreover, the selection of examples allowed identification of the limiting solution derived. Discussion is also included on the application of the model to industrial FB processes, focussing on the understanding of the model limitations in order to provide guidelines for extensions. This work complements existing generalised FB reactor models for catalytic gas–solid reactions, therefore, allowing similar generalised analysis for non-catalytic gas–solid reactions. Acknowledgments The authors acknowledge the European Commission, the Commission of Science and Technology of Spain, and Junta de Andalusia for their financial support. The first author acknowledges the Thermal and Fluid Engineering Department at University Carlos III (Madrid) for kindly invited him to lecture 16 part of the material of this article. Discussions from the lecturing sessions made, hopefully, the actual version of the paper more comprehensive and accessible. Appendix A. Frank-Kamenetskii’s approximation Frank-Kamenetskii [45] proposed an explicit solution for the variable yin a general equation of the form y−(1 −μy)n=0 with 0 <n<1 (A.1) where μis a constant of any value in the interval (0–2.7). This solution is y=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ min 1,1 μwith n=0 [({(1 −n)μ}1/n+1)n+nμ]−1with 0 <n<1 2n[(2n)1/n−1+(1 +2nμ)1/n +1]−nwith 1 <n<2.7 (A.2) Eqs. (21) and (29) can be expressed in the form of Eq. (A.1) where yis ηph or ηe, and the corresponding values of μare DaR/Naand DaRηi. This leads to explicit solutions given by Eqs. (26) and (31) for ηph and ηe, respectively. Appendix B. Formulation of the population balance: derivation of Eq. (38) For bed mass and flowrate of solids, as shown in Fig. 1b, an overall steady-state mass balance in the bed gives F0=rc,b+F1(B.1) rc,b is defined in Eq. (7). Now, by making a balance on conversion of the particles between xcand xc+dxcwe have wbKr d(F(xc)pb(xc)) dxc=F0p0(xc)−F1p1(xc) −wbR(xc)pb(xc) (B.2) In a well mixed bed pb(xc)=p1(xc). Solving for the distribution of conversion in the bed pb(xc) yields [10]: dpb(xc) dxc+dlnF(xc) dxc+1 λF(xc)+1 1/Yc0 −xcpb(xc) =1 Das p0(xc) F(xc)(B.3) where Dasand λare dimensionless parameters defined in Eqs. (40) and (41). Eq. (B.3) is integrated using the condition pb(0)=0 to obtain pb(xc)=1 Das 1/Yc0 −xc F(xc)xc 0exp −Θ(s) λp0(s) 1/Yc0 −sds (B.4) Assuming that all the particles enter with the same conversion xc0, the feed distribution is p0(xc)=δ(xc−xc0) and Eq. (B.4) is simplified to give Eq. (38). References [1] W.G. May, Fluidized bed reactors studies, Chem. Eng. Prog. 55 (1959) 49–56. [2] J.R. Grace, Fluid beds as chemical reactors, in: D. Geldart (Ed.), Gas Fluidization Technology, John Wiley & Sons Ltd., 1986, pp. 285–339. [3] J.C. Orcutt, J.F. Davidson, R.L. Pigford, Reaction time distributions in fluidized bed reactors, Chem. Eng. Prog. Symp. Ser. 59 (1962) 1–15. [4] J.F. Davidson, D. Harrison, Fluidised Particles, Cambridge University Press, London, 1963. [5] M. Horio, C.Y. Wen, An assessment of fluidized-bed modelling, AIChE Symp. Ser. 73 (1977) 9–21. [6] W.P.M. van Swaaij, Chemical reactors, in: J.F. Davidson, R. Clift, D. Harrison (Eds.), Fluidization, 2nd edition, Academic Press, London, 1985, pp. 595–629, Chapter 18. [7] J.F. Davidson, D. Harrison, R.C. Darton, R.D. LaNauze, The two-phase theory of fluidization and its application to chemical reactors, in: L. Lapidus, N.R. Amundson (Eds.), Chemical Reactor Theory, A Review, Prentice- Hall, Englewood Cliffs, NJ, 1977, pp. 583–685. [8] C. Chavarie, J.R. Grace, Performance analysis of a fluidized bed reactor. II. Observed reactor behavior compared with simple two-phase models, Ind. Eng. Chem. Fund. 14 (1975) 79–86. [9] H.S. Caram, N.R. Amundson, Fluidized bed gasification reactor modeling. 1. Model description and numerical results for a single bed, Ind. Eng. Chem. Proc. Des. Dev. 18 (1978) 80–96. [10] H.S. Caram, N.R. Amundson, Fluidized bed gasification reactor modeling. 2. Effect of the residence time distribution and mixing of particles. Staged bed modelling, Ind. Eng. Chem. Proc. Des. Dev. 18 (1978) 96– 102. [11] N. Mostoufi, H.P. Cui, J. Chaouki, A comparison of two- and single-phase models for fluidized bed reactors, 2001, Ind. Eng. Chem. Res. 40 (2001) 5526–5532. [12] D. Kunii, O. Levenspiel, Fluidization Engineering, 2nd edition, Butterworth-Heinemann, London, 1991. [13] R.K. Irani, B.D. Kulkarni, L.K. Doraiswamy, Analysis of complex reaction schemes in a fluidized bed. Application of the Kunii-Levenspiel model, Ind. Eng. Chem. Proc. Des. Dev. 19 (1980) 24–30. [14] T.P. Chen, S.C. Saxena, Mathematical modelling of coal combustion in fluidized beds with sulphur-emission control by limestone or dolomite, Fuel 56 (1977) 401–413. [15] B.W. Overturf, Toward a general description of gas–solid reactions in a fluidised bed reactor, Ph.D. Dissertation, University of Purdue, 1980. [16] J.G. Yates, Fundamentals of Fluidized-bed Chemical Processes, Butterworths, London, 1983. [17] L.K. Doraiswamy, M.M. Sharma, Heterogeneous Reactions. Analysis, Examples and Reactor Design, vol. 1, Wiley, New York, 1984. [18] A.B.M. Heesink, J. Klaus, W.P.M. Van Swaaij, Influence of particle residence time distribution on the reactivity in fluidized bed reactors, Chem. Eng. Sci. 49 (1994) 2243–2261. [19] L. Liliedahl, K. Sj¨ ostr¨ om, Modelling of char-gas reaction kinetics, Fuel 76 (1997) 29–37. [20] S. Dutta, C.Y. Wen, R.J. Belt, Reactivity of coal and char. 1. In carbon dioxide atmosphere, Ind. Eng. Chem. Proc. Des. Dev. 16 (1977) 20–30. [21] H. Huang, A.P. Watkinson, Coal gasification in a stirred bed reactor, Fuel 75 (1997) 1617–1674. [22] T. Adschiri, T. Shiraha, T. Kojima, T. Furusawa, Prediction of CO2gasifi- cation rate of char in fluidized-bed gasifier, Fuel 65 (1986) 1688–1693. [23] J. Szekely, J.W. Evans, H.Y. Sohn, Gas–Solid Reactions, Academic Press, New York, 1976. [24] F.G. van den Aarsen, Fluidised bed wood gasifier. Performance and modelling, Ph.D. Dissertation, Twente University Publication, University of Twente, 1985. [25] S.K. Bhatia, D.D. Perlmutter, A random pore model for fluid-solid reactions. I. Isothermal, kinetic control, AIChE J. 26 (1980) 379–386. [26] G.A. Simons, The unified coal-char reaction, Fuel 59 (1980) 143–144. [27] J.L. Johnson, Kinetics of Coal Gasification, Wiley, New York, 1979. [28] N.C. Gardner, E. Samuels, K. Wilks, Catalysed hydrogasification of coal chars, Adv. Chem. Ser. 131 (1974) 217–236. 17 [29] E. Chornet, J.M. Baldoroso, H.T. Tarki, Kinetic expressions for coal chargas reactions, Fuel 58 (1979) 395–397. [30] S. Kasaoka, Y. Sakata, C. Tong, Kinetic evaluation of the reactivity of various coal chars for gasification with carbon dioxide in comparison with steam, Int. Chem. Eng. 25 (1985) 160–175. [31] N.M. Laurendeau, Heterogeneous kinetics of coal char gasification and combustion, Prog. Energy Combust. Sci. 4 (1978) 221–270. [32] A. G´ omez-Barea, P. Ollero, R. Arjona, Reaction-diffusion model of TGA gasification experiments for estimating diffusional effects, Fuel 84 (2005) 1695–1704. [33] A. G´ omez-Barea, P. Ollero, B. Leckner, Mass transport effects during measurements of gas– solid reaction kinetics in a fluidised bed, Chem. Eng. Sci. 62 (2007) 1477–1493. [34] S.O. Oka, Fluidised Bed Combustion, Marcel Dekker Inc., New York, 2004. [35] G.F. Froment, K.B. Bischoff, Chemical Reactor Analysis and Design, 2nd edition, Wiley, New York, 1990. [36] O. Levenspiel, D. Kunii, T. Fitzgerald, The processing of solids of changing size in bubbling fluidized beds, Powder Technol. 2 (1968) 87–96. [37] G. Marb´ an, A.B. Fuertes, J.J. Pis, Calculation of burnoff distributions of char particles in a bubbling fluidized bed reactor by means of batch combustion experiments, Chem. Eng. Res. Des. 76 (1998) 864–872. [38] B. Srinivas, N.R. Amundson, A single-particle char gasification model, AIChE J. 26 (1980) 487–496. [39] J.I. Morell, N.R. Amundson, S.K. Park, Dynamics of a single particle during char gasification, Chem. Eng. Sci. 45 (1990) 387–401. [40] C.Y. Wen, Noncatalytic heterogeneous solid fluid reactions, Ind. Eng. Chem. 60 (1968) 34–54. [41] G. Brem, J.J.H. Brouwers, Analytical solutions for non-linear conversion of a porous solid particle in a gas. I. Isothermal conversion, Chem. Eng. Sci. 45 (1990) 1905–1913. [42] A. Gomez-Barea, B. Leckner, P. Ollero, C. Fernandez-Baco, L. Salvador, Reduction of physical effects during reactivity tests in fluidized bed, Ind. Eng. Chem. Res. 45 (2006) 7344–7350. [43] A. G´ omez-Barea, Modelling of diffusional effects during gasification of biomass char particles in fluidised-bed, Ph.D. Dissertation, University of Seville, Spain, 2006. [44] A. G´ omez-Barea, B. Leckner, M. Campoy, Conversion of char in CFB gasifiers, in: E.-U. Hartge (Ed.), Proceedings of 9th International Conference on Circulating Fluidized Beds, Hamburg, Germany, 2008. [45] D.A. Frank-Kamenetskii, Diffusion and Heat Exchange in Chemical Kinetics, Princeton University Press, Princeton, 1955. 18