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The Chemical Engineeri ng Journ al, 53 (1993) 25-37 25

A grain size distribution model for non-catalytic gas-solid reactionsA.B.M. Heesink, W. Prim and W.P.M. van SwaajjDepartment of Chemical Engin eeri ng, Tumate Uni vmsitp of Technology, PO Box 217, NL -7500 AE Enschede~ethm-lamis)(Received J anuary 8, 1993; in final form J une 4, 1993)

AbstractA new model to describe the non-catalytic conversion of a solid by a reactant gas is proposed. This so-called grain size distribution (GSD) model presumes the porous particle to be a collection of grains ofvarious sizes. The size distribution of the grams is derived from mercury porosimetry measurements. Themeasured pore size distribution is converted into a grain size distribution through a so-called pore-to-sphere factor whose value is also derived from the porosimetry measurements. The grains are dividedinto a number of size classes. For each class the conversion rate is calculated either according to theshrinking core model, involving core reaction and product layer diffusion as rate-determining steps, oraccording to a new model in which some reaction at the grain surface is assumed to be limiting. TheGSD model accounts for the phenomenon of pore blocking by calculating the maximum attainableconversion degree for each size class. In order to verify the model, two types of precalcmed limestoneparticles with quite different microstructures were sulphided as well as sulphated. Furthermore, a singlesample of sulphided dolomite was regenerated with a mixture of carbon dioxide and steam. For eachreaction good agreement was attained between measured and simulated conversion vs. time behaviour.

1. IntroductionGas-solid reactions are involved in many industrial

processes. These reactions are generally of the typec&sol) + bB(gas) - cC(so1) + dD(gas)Some examples are the removal of sulphur dioxide(hydrogen sulphide) by a solid sorbent in hightemperature flue gas (coal gas) cleaning, the roastingof iron sulphide and the subsequent reduction ofiron oxide to produce iron in steelworks and thecalcination of limestone in the cement industry.The microstructure of a porous solid affects itsreactivity towards a reactant gas. Both dilIusivityand kinetics depend on the microstructure, whichmay change during conversion. For instance, whenthe molar volume of the solid product exceeds thatof the solid reactant, pore blocking might occur,resulting in zero diffusivity and cessation of con-version. Mathematical models for gas-solid reactionsshould include the influence of (changing) micro-structure on particle diffusivity and volumetric ki-netics. The term volumetric kinetics refers to theproduct of specific surface area and intrinsic kinetics,the latter reflecting the combined result of surfacereaction kinetics and product layer diffusion. Many

mathematical models have been developed. Theycan be roughly divided into two categories, namelygrain models and pore models.

According to the gram model concept, a poroussolid consists of small impervious pieces or grainsthat are dispersed in gas, which is then the con-tinuous phase. Each grain is assumed to be convertedaccording to the shrinking core model developedby Yagi and Kunii [ 11. This concept was elaboratedby Szekely and Evans [2], who derived relations tocalculate the conversion VS. time behaviour of solidsconsisting of spherical, cylindrical or slab-like grams.The model of Szekely and Evans does not accountfor the change in grain size which occurs when themolar volumes of the solid reactant and solid productdiffer. Moreover, it is assumed that all grains areof equal size. Hartman and Coughlin (31 modifiedthis model to simulate the sulphation of calcinedlimestone. They actually took into account the ex-pansion of the calcium oxide grains during con-version. Later, Georgakis et a l . [ 4 ] presented asimilar changing grain size model. In these modelsall grains are still assumed to have initially the samesize. Szekely and Propster [5] studied the effect ofgrain size distribution on the conversion behaviourof porous solids. Still assuming that grains do not

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26 A.B.M. Heesink et al. / GSD model for non-catalytic gas-solid reactionschange in size during conversion, they illustratedthat a distribution in grain size greatly affects theparticle conversion behaviour. Szekely and Propsterregarded some statistical distributions and closedtheir publication with the statement it would appearthat when precise interpretation of the (conversionbehaviour) measurements is required, in terms ofthe grain model, it would be desirable to complementthe usual kinetic and structural measurements witha determination of the grain size distribution. Morerecently, Dam-Johansen et al. [6] developed agrain-microgram model to describe the sulphationof calcium oxide. According to their model, thesolid consists of porous grains which in turn arecomposed of impervious micrograms. The micro-grains are converted by a shrinking, unreacted coremechanism until all free space inside a grain hasbeen filled with product. Then the grains start toreact according to the shrinking, partially reactedcore mechanism. In this model both grains andmicrograins are assumed to have initially the samesize. Although the particle conversion behaviour canoften be well described with an appropriate grainmodel, the dependence of surface area upon con-version is not well predicted [ 71. Therefore Efthim-iadis and Sotirchos [81 developed the partially over-lapping grain model in which each grain is assumedto consist of a hard core, not overlapping with thecores of neighbouring grains, and a soft permeableshell which can overlap with the shells of othergrains. By adjusting the ratio of core radius andgrain radius, the development of surface area duringconversion can be simulated.In contrast with grain models, pore models regardthe solid phase as the continuous phase. Variouspore models have been developed, each describingpore tortuosity, pore interconnectivity and pore sizedistribution in its own way. Ramachandran and Smith[9] developed a single-pore model which describespore diffusion, diffusion through a product layerbeing precipitated on the pore walls, and surfacereaction within a single cylindrical pore. The phe-nomenon of pore mouth closure is included as well.Ramachandran and Smith assumed all pores to haveinitially the same size. Later models took into accountthe distribution in pore size [lo-l 31 and the in-terconnectivity of pores [ 141. Good results havebeen obtained with these more complex modelswhere the influence of pore size distribution wasclearly shown (e.g. by Sotirchos and Zarkanitis [ 151).A new model is now proposed. This grain sizedistribution (GSD) model is based on the grainmodel of Szekely and Evans [21, but additionallyaccounts for a measurable distribution in grain size,changing grain size and the possibility of pore

blocking. This is done by converting the measuredpore size distribution into a grain size distributionthrough a newly defined pore-to-sphere factor.Pore blocking is assumed to occur when the ex-pansion of grains from a certain grain size classequals the volume of pores from the correspondingpore size class. The GSD model is meant to describethe volumetric kinetics of solid particles while ac-counting for the effects of pore-grain size distri-bution and pore blocking. Unlike in (partially) over-lapping grain models, the grams are assumed to beunconsolidated and non-overlapping. Thus, althoughthe evolution of volumetric kinetics during con-version can be well described, the GSD model willnot predict the evolution of surface area duringconversion very well. Furthermore, intraparticletransport is not considered: it is assumed that thegas phase inside a particle is homogeneous in nature.The model should therefore only be applied toparticles which are converted in the kineticallycontrolled regime. However, it can be modified inthis respect by combination with a proper poremodel. This will be illustrated in a future paper.

It is important to note that the present modelcannot be used to describe the conversion behaviourof particles in which fresh pores (and grains) developduring conversion. However, when the formationof pores (and grains) only takes place during theinitial stage of conversion (nucleation stage), theGSD model may be applied to describe conversionvs. time behaviour after the nucleation stage. Thisapproach is similar to that of the crackling grainmodel developed by Park and Levenspiel [ 161.

First, the shrinking core model will be discussed.This classical model, which discriminates betweenproduct layer diffusion and reaction at the surfaceof the unreacted core (core reaction) as possiblerate-determining steps, will be extended by a thirdstep, namely reaction at the surface of the grain(grain reaction). Second, it will be shown how thegrain size distribution is derived from a mercuryporosigram by applying the concept of a pore-to-sphere factor. Finally, the GSD model will be verifiedexperimentally. The calculated conversion vs. timebehaviour will be compared with the measuredbehaviour during the sulphation and sulphidationof two types of precalcined limestone particles withdifferent microstructures. This will also be done forthe regeneration of sulphided dolomite particleswith a mixture of steam and carbon dioxide. TheGSD model is used to establish the governing mech-anisms of the reactions involved.


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A.B.M. Heesink et al. / GSD model f or non-catc&tic gas-solid reactions 272. Sh r i n k i n g core model

The classical model is illustrated in Fig. 1, wherea sketch of a partly converted sphere is linked upwith a diagram showing the corresponding radialprollle of the involved gaseous reactant. The outershell of the spherical particle consists of solidproduct only, whereas the core contains the originalsolid reactant. Core and product layer materials donot mix, implying that there is a sharp interfacebetween core and product layer. Two different con-centration profiles are shown, one referring to theextreme case of conversion fully controlled by corereaction, the other to the extreme case of conversionfully controlled by product layer diffusion. Below,each extreme is worked out. Afterwards, a thirdpossible limiting step, namely grair? reaction, isconsidered.2.1. Core r e a c t i o n l im i t a t i o nIn the case of core reaction limitation the con-version rate of a grain with initial radius R, can becalculated from

R , represents the radius of the unconverted core,which can be calculated according toR,=(l -X) lR,, (21The parameter k , represents the kinetic constantof the surface reaction at the core, which is assumedto be of ilrst order in the gaseous reactant. C,represents the concentration of the gaseous reactantat the core surface, which in the case of core reactionlimitation equals the concentration at the grainsurface, C,. The parameter N, represents the initialconcentration of the solid reactant in the grain anddepends on the purity of the grain (P) and themolecular volume of the solid reactant:

Concentration at Concentration atgrain border

Product-layer diffusionlimitation

Fig. 1. Illustration of the shrinking core model for the conversionof a single grain. Concentration profiles: left-hand side, corereaction limitation; right-hand side, product layer diffusion lim-itation.

PNo= ~V ol eac (3)It is assumed here that the density of impurities inthe grain is equal to that of the solid reactant.Combination of eqns. (1) and (2), while substi-tuting C, by C,, yieldsdx 3X,( 1-X) for X


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28 A.B.M. Heesink et al. / GSD model f or rwn-catalytic gas-solid reactions

I 0 for X=X,,(IO)The corresponding overall reaction rate constant(in the case of product layer diffusion control) Kp

is then(11)

The overall reaction rate constants K , and Kp a sgiven by eqns. (5) and (11) respectively appear tobe interrelated according to

(12)It should be noted that eqn. (10) cannot be used

to calculate the initial conversion VS. time behaviour,i.e. when no product layer has been built up yet.Use of eqn. (10) would yield an infinitely high initialconversion rate. Here it is assumed that the corereaction limits the initial conversion rate. Thereforea general expression including both core reactionand product layer diffusion as possible steps isderived below.

2.3. General ewessicm a c co r d i n g t o t h ec l a ssi c a l sh r i n k i ng cor e mode lCombining eqns. (1) and (7), while substitutingR , and R, according to eqns. (2) and (S), andintroducing K , and $ as defined by eqns. (5) and(12) results in a general expression for the con-version rate from which C, has been eliminated.This expression is valid for both core reaction controland product layer diffusion control as well as forintermediate cases.dX-=dt

eqn. (13) turns into eqn. (4) which is valid in thecase of core reaction control. The value of rC,husdetermines which mechanism is rate controlling:rj +z 1: core reaction limitationrF,+ 1: product layer diffusion limitation (14)Note that eqn. (13) meets the demand of a limitedinitial conversion rate:

(15)The time needed to obtain a certain conversion canbe derived by integration of eqn. (13):

l-(l-X)3 I (6tQ= K c =&x l-(l-X)zB+ ;[l-(KX+l)ZB]( 1

for X


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A.B.M. Heesink et al. / GSD model for rwn-catal@c gas-solid reactions 29

X(t) = n cm-Lax for x=x_

From eqn. (18) it follows that the conversionrate increases as conversion proceeds and actuallyreaches its maximum value at complete conversion,i.e. when the grain has its highest surface area.Although the phenomenon of an increasing con-version rate during the first part of a conversionprocess has been observed in practice (e.g. by Tsengand Edgar [ 17 1 during the combustion of coal char),such behaviour has never been observed during thefinal part of conversion. This probably explains thefact that grain reaction limitation is not consideredin the classical shrinking core model. However, aswill be shown below, grain reaction limitation canlead to normal conversion VS. time behaviourwhen the grain size distribution is taken into account.Therefore grain reaction limitation is consideredhere as a third possible mechanism.It is important to note that it is not possible tocouple eqns. (13) and (18) by elimination of C,The parameter C, used in eqn. (13) (by means ofK,) refers to the concentration of the (ionic) specieswhich is formed at the grain surface, diffuses throughthe product layer and subsequently reacts at thecore surface. The parameter C, in eqn. (1 S) (through&Q refers to the concentration of the adsorbedgaseous reactant at the grain surface. In general itis not possible to combine the three steps of grainreaction, product layer diffusion and core reactionwithout violating the assumption of the classicalshrinking core model that the concentration of thegaseous reactant (Le. ionic species) at the grainsurface remains constant during conversion. There-fore, when simulating measured conversion VS. imebehaviour with the GSD model, we will only applythe three extreme cases of grain reaction limitation(eqn. (20)), product layer diffusion limitation (eqn.(16) with $i=*-1) and core reaction limitation (eqn.(16) with J/=0). Intermediate cases will not beconsidered in order to avoid violation of the classicalshrinking core model and to reduce the number ofmodel fit parameters (k,, k, and Q) applied.

3. Grain size distribution modelAccording to the GSD model, a porous particleis regarded as a collection of small spherical grainsof various sixes. Figure 2 illustrates this conceptfor a calcined limestone particle, which has a bi-disperse nature according to the literature [6, 181.

Fig. 2. Representation of the GSD model concept for a partlyconverted bidisperse particle: M, macropores between clusters;m, micropores within clusters, between grains.

The grains are gathered in clusters. Micropores arelocated within these clusters whereas macroporesare located in between the clusters. It should benoted that the GSD model is not only applicableto bidisperse particles. Basically, the conversion vs.time behaviour of all types of particles, includingof course simple monodisperse ones, can be de-scribed.Frevel and Kressley (191 studied mercury po-rosigrams of dense packings of microspheres withradii between 120 and 180 pm. They concludedthat the apparent pore size within these packingsis determined by the size of the spheres only. Fora packing of uniform microspheres a value of threeis found for the ratio of sphere radius and poreradius. This ratio is referred to as the pore-to-sphere factor F. For a mixture of different sizespheres the porosimetry curve obtained appears tobe very similar to a smoothed composite of theporosimetry curves measured for grains from theindividual size classes. In this case the value of thepore-to-sphere factor depends on the size distri-bution of the spheres and the mode of packing butis always smaller than three.In the present work the findings of Frevel andKressley [ 191 are used to derive the sizes of thegrains inside a porous solid from mercury poro-simetric measurements. To verify whether this isallowed for spheres with sizes typical for grainsinside a porous solid (nanometre range), mercuryporosimetry was performed on a test sample ofcalcined limestone particles which were also studiedby means of scanning electron microscopy (SEM).Figure 3(a) and 3(b) show respectively the mercuryporosigram and the SEM photograph on whichspherical grains of various sizes can clearly bedistinguished.The measuredpore radiusranges fromabout 7 to 40 nm whereas the average pore radiusamounts to about 17 run. The radii of some 70grains were determined. The radii range from about


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A.B.M. Heesink et aL / GSD rnoakl f or non-catalytic gas-solid reactions

!bl

Fig. 3. Verification of the relationship between gram radius andpore radius by comparing (a) a mercury porosigram and (b)an SEM photograph of a test sample of calcined limestoneparticles. Value of pore-to-sphere factor used in (c) is 1.8. Poresize classes in (c): 1, 6-S; 2, 9-12; 3, 13-15; 4, 16-18; 5,19-21; 6, 22-24; 7, 25-27; 8, 28-30; 9, 3133; 10, 34-36nm.

12 to 40 run, the average grain radius being some30 nm. When comparing the average pore radiuswith the average grain radius, a pore-to-sphere factorof 30/l 7 = 1.8 can be calculated. After dividing themeasured grain radii by the value of 1.8, both poresand grains were divided into 10 size classes. In Fig.3(c) the volume fractions of the various pore andgrain classes are plotted. The agreement betweenpore and grain size distributions justifies the con-clusion that a grain size distribution can indeed bederived from mercury porosimetric measurements.

-._--0 30 60 90 120 150

Pore radius (MI)It will now be shown how the actual value of the Fig. 4. Division of a mercury porosigram into intervals (herepore-to-sphere factor can be derived from a mercury 10) according to the GSD model.

porosigram. The mercury porosigram is first dividedinto a limited number (N) of pore radius intervals.Figure 4 shows how this is done. Since particlereactivity is proportional to the amount of specificsurface area involved, a criterion is available fordeciding which part of the porosigram should betaken into account. In the case of Fig. 4 pores witha radius larger than about 110 nm (representingonly a minor part of the total specific surface area)are neglected. These pores are considered to bemacropores which are located between clusters ofgrains. Therefore the sizes of the macropores areassumed to correspond to the sizes of clusters ratherthan grains. According to the definition of the pore-to-sphere factor, the grain radius that correspondsto the pore radius of a certain interval is given byKS i =FRp, i (211The pore volume for interval i, Vp, i , is a measureof the total weight of grains with radius R ,,, . Theweight fraction of grains belonging to size class i,Vi, GUI thus be calculated from

(2%

The specific surface area A of particles containingdifferent size spheres can be obtained fromA= (33)Thus, once the specific surface area of the solidreactant is known, the value of the pore-to-spherefactor can be derived from

The value of A can be measured in several ways,but also by mercury porosimetry. Consequently, a


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A.B.M. Heesinlc et aL / GSD model f or non-catalytic gas-solid reactions 31single mercury porosimetry measurement is suffi-cient to determine the value of the pore-to-spherefactor and subsequently the grain size distribution.With the known values for R,,i the conversionof each grain class, Xi, can be calculated as afunction of time using either (the inverse of) eqn.(16) when core reaction or product layer diffusionis the governing mechanism or eqn. (20) in thecase of grain reaction limitation. In the former casecalculations have to be performed numerically, sinceeqn. (16) cannot be inverted in an analytical way.For proper use of eqns. (16) and (20) the maximumattainable conversion of each grain class, X_,+,must be known. In our model VP,+ s assumed torepresent the volume available for the expansionof grams from size class i. Then the value of X,,, ican be calculated from VP , and the expansion factorK, which is a measure of the difference betweenthe molar volumes of solid reactant and solid prod-uct:xmax,i= Pso1eac P*K vi (if < 1, else X,,, i = 1) (25)An important assumption of our model is that theaccess to grains of a certain size class is not disturbedwhen grains of other size classes have reachedmaxirmmr conversion and their corresponding poreshave become blocked.Finally, the total conversion of a particle is givenbyx,, = i$ixd (26)i-l

4. Experimental detailsThe GSD model has been tested by comparingseveral experimental ly determined conversion vs.time curves with predicted ones. All conversionexperiments were carried out by measuring thechange in sample weight during reaction using

thermogravhnetric (TG) analysis. Three reactionsof importance in the regenerative high temperaturedesulphurization process being developed at ouruniversity were investigated.1. Sulphidation of precalcined limestone:CaO + HaS - CaS + Hz0

2. Sulphation of precalcined limestone:CaO + SOa + #, - &SO,

3. Regeneration of sulphided dolomite:CaSMgO + COa + Ha0 - CaCO,.MgO -t HaS

4.1. Su l p h k i a d o n a n d su l p h a t i o nSulphidation and sulphation experiments werecarried out in an atmospheric thermogravimetricanalyser (TGA, type Setaram TG-85) using two typesof limestone. The first (Wiilfrath) is a natural lime-stone which did not undergo any pretreatment. Thesecond (Lhoist agglomerate) consists of particleswhich were agglomerated from limestone powder(75 wt.% of powder particles smaller than 75 pm,25 wt.% smaller than 150 pm) by applying 2 wt.%bentonite cement. Table 1 gives the compositionsof the materials used.A sample of about 1.5 g (particle diameters be-tween 106 and 212 pm) of each limestone typewas calcined in the TGA at 800 C while fhrshingwith helium:CaCOa - CaO + COaAfter about 15 min no further decrease in weightwas observed and Cal&nation was stopped. About1 g of the calcined material was set aside for mercuryporosimetry measurements (porosimeter Carlo ErbaStrumentazione, DRU model 204). The remainderwas used for the sulphidation and sulphation ex-periments.Figure 5 shows that the porosigram of the calcinedLhoist agglomerate particles has two peaks, whereasonly one peak has been found for the Wiil frathparticles. Because of this pronounced difference inmicrostructure, the two selected particle types arequite suitable to test the GSD model. The specificsurface area derived from the mercury porosimetrymeasurements amounts to 21.5 m2 g- for theWiil frath particles and 25.3 m2 g-r for the Lhoistagglomerate particles. Most of the surface area ofthe Wiilfrath particles corresponds to pores smallerthan 100 run. Accordingly, pores with a radius largerTABLE 1. Chemical compositions (wt.%) of the limestone anddolomite particles used for experimental verification of the GSDmodel

Component

CaC03MgCO3SiFeAlsKSrNaCaO contentafter calcination

Wiilfrathnatural

97.100.900.800.17


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32 A.B.M. Heesink et al. / GSD model f or non-catalytic gas-solid reactions

110 100 1000 1 10 100 1000(a) Pore radius (nm) @I Pore radius (MI)Fig. 5. Mercury porosigrams of calcined limestone partic les: (a) Wiilfrath; (b) Lhoist agglomerate.than 100 nm are considered as macropores locatedbetween clusters of grams. For the Lhoist agglom-erate particles this boundary between micro- andmacropores is located at approximately 200 nm.

4.2. Ad regene ra t i on

The observed weight change during the TGAexperiments should only reflect the reaction kineticsof the sample. In order to minimize the influenceof mass and heat exchange between the samplebasket and gas atmosphere, the amount of sampleused for the sulphidation and sulphation experimentswas kept small (approximately 3 mg). Differentialoperation was assured by the combined use of asmall amount of sample and a sufficiently high gasflow rate through the TGA, i.e. 400 ml (STP) mm-.

Prior to regeneration, about 600 mg of Lhoistdolomite particles (diameters between 106 and 212pm) were fully calcined in a high pressure TGA(HP-TGA, constructed by DMT). Table 1 gives thecomposition of the applied Lhoist dolomite particles.Calcination was performed under nitrogen at a tem-perature of 700 C and at atmospheric pressure:CaC03.MgC03- CaO.MgO + 2COZ

The sulphation reaction was carried out at 700C. The feed gas mixture contained 0.2 vol.% sulphurdioxide, 9.8 vol.% nitrogen, 2 vol.% oxygen andbalance helium. Sulphidation experiments were alsocarried out at 700 C while using a gas mixture of2 vol.% hydrogen sulphide, 4 vol.% hydrogen (tostabilize the hydrogen sulphide) and balance helium.Before adding the hydrogen sulphide to the gasflow, the hydrogen-helium mixture was passedthrough a bed (heated to 85 C) of oxygen-bindingcopper catalyst (type BASF R 3-11) to remove smallquantities of oxygen originally present in the bottlegas. This was done because experiments had shownthat even the presence of 5 ppm oxygen by volumein the gas mixture leads to significant oxidation ofthe calcium sulphide product towards calcium sul-phate. The formation of calcium sulphate results inmisinterpretation of the measured conversion vs.time behaviour as a consequence of the large dif-ference between the molecular weights of calciumsulphate (Jf= 136) and calcium sulphide (M= 72).

Cal&ration was stopped after about 4 h when nofurther decrease in weight was observed. About 300mg of the obtained CaO.MgO particles were usedfor mercury porosimetry. Figure 6 shows the ob-tamed porosigram. This porosigram was used in-directly to obtain the size distribution of the CaSMgOgrains. This method was chosen because directmercury porosimetry of the sulphided particles hadbeen shown to be disturbed by the fast oxidationof CaSMgO towards CaSO,.MgO occurring whenthe CaSMgO sample was taken out of the HP-TGAand contacted with air. First the size distributionof the CaO.MgO grams was derived in the waydescribed above using the porosigram of Fig. 6.The boundary between micro- and macropores was

All required gases were taken from bottles. The 1 10 100 1000composition of the applied gas mixtures was con- Port radius (nm)trolled by means of calibrated electronic mass flow Fig. 6. Mercury porosigram of folly calcined dolomite particlescontrollers. (Lhoist dolomite).


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A.B.M. Heesink et al. / GSD model f or rum-catalytic gas-solid reactions 33fixed at 120 run. Then the obtained distribution wastransformed into the size distribution of the CaSMgCgrains by use of eqn. (8) (K=O.38).The remaining 25 mg of fully calcined dolomitenot used for mercury porosimetry were sulphidedin the HP-TGA at 600 C and atmospheric pressurewhile using a gas mixture consisting of 2 vol.%hydrogen sulphide, 18 vol.% hydrogen and balancenitrogen:CaO.MgO + HaS - CaS.MgO + Ha0The required gases were taken from bottles. Anytraces of oxygen were removed from the nitrogenstream in the same way as described above for theatmospheric TGA. Sulphidation was stopped aftersome 30 ruin when no further change in weightwas observed.After sulphidation was completed the conditionswere adjusted for regeneration. During adjustmentthe sample was flushed with helium. Regenerationwas carried out at a temperature of 600 C and apressure of 10 bar. The applied gas mixture consistedof 20 vol.% carbon dioxide, 20 vol.% steam andbalance nitrogen. The total flow rate was fixed at5 1 (STP) mm-. The steam was obtained by evap-orating a small stream of water which was suppliedby an HPLC pump. Again, any traces of oxygenwere previously removed from the nitrogen stream.The composition of the applied gas mixtures wascontrolled by means of calibrated electronic massflow controllers.

6. Results and discussionIn this section the results of TGA measurementswill be compared with GSD model predictions. Ex-perimentally determined conversion vs. time curveswill be simulated by model computations in whicheither product layer diffusion, core reaction or grainreaction is assumed to be rate controlling. Inter-mediate situations will not be considered.As explained above, use of the GSD model requiresa mercury porosigram to calculate the pore-to-spherefactor and subsequently the size distribution of thegrains. Figures 5 and 6 have been used for thepresent calculations. The Wiil frath porosigram wasdivided into 36, the Lhoist porosigram into 53 andthe Lhoist dolomite porosigram into 47 pore sizeclasses. The largest pore radii involved are 125 run(Wiilfrath), 203 run (Lhoist agglomerate) and 123nm @hoist dolomite). Larger pores do not representa significant surface area and are regarded as ma-cropores. Values of 1.88 (Wiilfrath), 1.04 (Lhoistagglomerate) and 1.34 (Lhoist dolomite) have been

derived for the pore-to-sphere factors. As alreadystated above, the grain size distribution derived fromthe porosigram of the fully calcined Lhoist dolomiteparticles (Fig. 6) was transformed into the corre-sponding grain size distribution of the sulphidedparticles by making use of eqn. (8) with a K valueof 0.38. Accordingly, the radii of the fully calcinedgrains were multiplied by 1.113 to obtain the cor-responding radii of the sulphided grains.In the following presentation of results, valuesof the model fit parameters k,, D, and k, havedimensions corresponding to volumetric dimensions(mol me3) of C,. In the case of sulphation andsulphidation C, refers to the volumetric (bulk) con-centrations of sulphur dioxide and hydrogen sul-phide respectively. Then the parameters k, and kghave dimensions of m s- and D, has dimensionsof m2 s- . In the case of regeneration C, refers tothe product of the volumetric (bulk) concentrationsof carbon dioxide and water. Then the parametersk, and k, correspondingly have dimensions of m4mol- s-i and D, has dimensions of m6 mol- s- .I t should be noted, however, that the GSD modelcan also be applied when an adsorption-desorptionequilibrium exists at the grain surface. In that caseC, would represent a surface concentration and k ,,k , and D, would have the corresponding dimensions.5.1. Su l p ha t i o n

F i g u r e 7 shows conversion 21s. ime diagrams inwhich the measured and calculated best-fit curvesfor the sulphation experiments are plotted. As aresult of the large difference between the molarvolumes of calcium oxide and calcium sulphate(1.68x lo- and 5.22~ lo- m3 mol- respec-tively), pore blocking occurs and total conversioncannot be achieved. The maximum possible sul-phation degree of Lhoist agglomerate particles (0.6)appears to be much larger than that of Wiilfrathparticles (O-4), which can be explained by theirdifference in microporosity [ZO] (see also Fig. 5).The GSD fits of both measured conversion vs.time curves are quite satisfactory in the case whereproduct layer diffusion is assumed to be the gov-erning mechanism. The GSD model is thus able todescribe the influence of microstructure on con-version vs. time behaviour quite well. The best-fitvalue for D, amounts to 2.6 x lo-i3 m2 s-l for bothlimestone types. When calculating the product layerdiiusion fits using eqn. (16)) the k , value was fixedat 1 m s- . The resulting J I values are 3 X lo4 orlarger (depending on grain size; see eqn. (12)),indicating that product layer diffusion is indeed thegoverning mechanism in these fits.The obtained best-fit value ofD, appears somewhatlow when compared with values used in previous


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34 A.B.M. Heesink et al. / GSD model for rwn-catalytic gas-solid reactions

0.6

0 300 600 900 1200 1500 0 1000 2000 3000 4000(aI Time (s) (b) Time (s)Fig. 7. Measured and calculated conversion VS. time curves for the sulphation of (a) WClfrath and (b) Lhoist agglomerate particles(D,=2.6x10-3 m2 s-, k,=5.3~10-~ m s-, k,=3.0~10-~ m s-).

modelling efforts. Marsh and Ulrichson [21 ] recordedvalues ranging from 6.0~ lo-l3 to lo- m2 s-l.These values were derived from experiments carriedout by several investigators at temperatures between650 and 980 C. Borgwardt and Bruce [22] derivedvalues of D, at various temperatures ranging from800 to 1125 C. They reported an activation energyof 153 kJ mol - . Using this number, our D, valueof2.6X10-3m2s-at700Cwouldyieldavalueof 3.3 X lo- m2 s- at 850 C, which is in betweenthe data measured by Hartman and Coughlin [3]((0.6-0.86) X lo-l2 m2 s-l) and those measured byBorgwardt [23] (15~10~~ m2 s-l).The fits for core reaction limitation and grainreaction limitation in Fig. 7 are less accurate. Forthe Wtilfrath particles reasonable fits can still beobtained using a k, value of 5.3~ 1O-5 m s-l (ata fixed D, value of 1013 m2 s-l, resulting in $ valuessmaller than 10-24, indicating strong core reactionlimitation) or a k, value of 3.0 X 10m5 respectively.However, if the same k, or k, value is used for theLhoist agglomerate particles, very bad fits are ob-tained. Therefore the conclusion that product layerdiffusion governs the sulphation rate seems to bejusti6ed. This conclusion is supported by the ex-periences of other researchers. A possible mech-anism in which the process of product layer diffusionis rate controlling was proposed by Borgwardt etal. 24].

1. SOa + 302 + SO3 (gas phase reaction).2. SOa + 02- + S042- (grain reaction).3. S042-/02- counterdiffusion, (product layer dif-fusion - rate controlling).4. S042- + CaO + &SO, + 02- (core reaction).

5.2. SulphidationFigure 8 shows that the Lhoist agglomerate aswell as the Wiilfrath particles can be sulphidedcompletely. Pore blocking does not occur because

of the relatively small difference between the molarvolumes of calcium oxide and calcium sulphide(1.68~ 10e5 and 2.76~ 10e5 m3 mol- respec-tively) .Roughly considered, reasonable fits have beenobtained for core reaction limitation and grain re-action limitation. However, upon close inspectionit seems that fitting based on the assumption ofcore reaction limitation (kc= 1.0 X 10e4 m s-l,$< 10-22) is slightly better, especially for the Wiil-frath particles. This would lead to the conclusionthat some core reaction controls the sulphidationrate at the applied temperature of 700 C.

Borgwardt et al. 25] lsonvestigated the mech-anism of the reaction between calcium oxide andhydrogen sulphide. They varied the specific surfacearea of calcined limestone particles by controlledsintering. In the grain theory this specific surfacearea is related to the initial grain radius (it is assumedthat all grains have the same radius):R,= 3Ap,,l, reac (27)According to this relationship, Borgwardt et al. [25]implicitly varied the initial grain radius R, by varyingthe specific surface area. Prom eqn. (16) it followsthat the time needed to reach a certain degree ofconversion is proportional to R, in the case of corereaction limitation and proportional to R,,2 in thecase of product layer diffusion limitation. Borgwardtet al. [25] found that the time needed to reach70% conversion is proportional to R,,2-3 and con-cluded that product layer diffusion governs thesulphidation rate. This conclusion is in contrast withthe indication of core-reaction-controlled sulphi-dation provided by the present study. A possibleexplanation for this difference may be that Borgwardtet al.251, while sintering their limestone particles,


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A.B.M . Heesin k et al. / GSD model far mm-catalytic gas-soli d reacti4m.s

1.00.80.60.40.20.0

35

0 30 60 90 120 150 0 30 60 90 120 150(a> Time (s) @) Time (s)Fig. 8. Measured and calculated conversion VS. time curves for the sulphidation of (a) Wiilfrath and (b) Lhoist agglomerateparticles (Ds= 1.0x lo- m2 smJ, kc= 1.0X 10m4 m s-, ks=6.OX low6 m s-).

not only changed the initial grain size but also thereactivity at the core surface. It is quite feasiblethat the value of k, decreases with increasing sin-tering time, e.g. because a more perfect (and there-fore less reactive) calcium oxide lattice structureis formed. This seems to be coniirmed by the factthat the sulphidation rate observed in this study isconsiderably higher than expected from the studyof Borgwardt et al. [25 . On assuming product layerdiffusion to be the rate-controlling mechanism, aD, value of 1.0X lo-i2 m2 s- is found by fittingthe present experimental results. This value is muchhigher than the one derived from the work ofBorgwardt et al. [25] (D,=3.6~10-~ m2 s-r at700 C).Borgwardt et al. [25] proposed the followingsulphidation mechanism.1. H2S+ 02- + Hz0 + S2- (grain reaction).2. S2-/02- counterdiffusion (product layer dif-fusion).3. S2- + CaO + CaS + 02- (core reaction).They suggest that product layer diiusion is ratecontrolling. According to our ilndings, it is morelikely that the core reaction is rate controlling.A future paper will deal in more detail with thekinetics and mechanism of calcium oxide sulphi-dation.5.3. Regenera t i onF i g u r e shows the measured and calculated best-fit conversion vs. time curves for the regenerationof sulphided Lhoist dolomite particles. Despite thelarge difference in the molar volumes of calciumsulphide and calcium carbonate (2.76 X 10m6and3.69 x 10e5 m3 mol- respectively), full conversioncan be obtained owing to the presence of inertmagnesium oxide.By far the best fit is obtained when grain reactionlimitation is assumed. The best-fit value of kg

1.21.00.80.60.40.20.0 r . I - 1 I I

0 200 400 600 800Time (s)

Fig. 9. Measured and calculated conversion vs. time curves forthe regeneration of sulphided Lhoist dolomite particles(D,= 1.0~ 10eJ7 m6 mol- s-l, kc= 1.5x 10-O m4 mol-J s-,k,=7.5X 1O-1o m4 mol- s-).amounts to 7.5X1O-o m4 mol- s-. Althoughsome studies were performed previously [26, 271,the precise mechanism of the regeneration reactionhas not yet been established. Huang et a l . [27]suggest that product layer diffusion might be ratecontrolling but do not present any evidence for this.A reaction mechanism in which the grain reactionis rate controlling might be as follows.1. CO2+ Hz0 + S2- + C032- + H2S (grain reaction- rate controlling).

2. C032-/S2- counterdiffusion (product layer dif-fusion).3. co33- + CaS CaC03 + S2- (core reaction).A future paper will deal with the kinetics andmechanism of the regeneration reaction of sulphideddolomite with mixtures of carbon dioxide and steamin more detail.5.4, Gra i n s i z e d i st r i b u t i o n vs. s i n g l e g r a i ns ize

It is interesting to see whether the effort of takingthe grain size distribution into account results in


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36 ABM . Heesink et al. / GS1) model for m-catalytic gc~+solicl reactionsa significantly better description of the conversion21s. ime behaviour. Therefore GSD fits were preparedwith a varying number of grain size intervals: one,two and the full number of intervals. Figure 10shows the results for the best-fit mechanisms ob-tained for sulphation, sulphidation and regeneration.The quality of the fits does indeed increase withincreasing number of grain size intervals, especiallyin the case of product layer d~usion and grainreaction limitation.

0.8 1 I

0.0-t0 1000 2000 3cm 4000

(a) Time (s)

(h)i0 30 60 90 120 150

Time (s)

l . O-0.8 -0.6 -0.4 -0.2 -

0 200 400 600 800Cc) Time (s)Fig. 10. Measured and calculated conversion vs. time curvesfor one, hvo and the full number of pore-grain size intervals[a) product layer diffusion fits for the sulphation of agglomeratedLhoist limestone @,=2.6x IO-l3 m2 s-l); (b) core reactionfits for the sulphidation of Wiilfrath limestone (Ic,= 1.0X IO-*m s-l); c) rain reaction fits for the regeneration of sulphidedLhoist dolomite (k,=7.5X10- m4 mol- s-)_

6. ConclusionsA gram size dist~bution model was developed todescribe the kinetics of non-catalytic reactions be-

tween a reactant gas and a porous solid. The evo-lution of the solid microstructure during conversionand its influence on reactivity is explicitly takeninto account. A measurable pore-to-sphere factoris used to transform the pore size dis~bution ob-tamed from mercury porosimetric measurementsinto a gram size distribution. The applicability ofthis method to micro grains was verified experi-mentally.The proposed model was tested by comparingthe calculated conversion DS. ime behaviour withthe measured behaviour during the sulphation andsulphidation of two types of limestone particles ofquite different microstructures as well as during theregeneration of sulphided dolomite. The model wasapplied successfully to determine the mechanismsof the gas-solid reactions involved. Under the ap-plied experimental conditions, sulphation appearedto be governed by product layer diffusion, sulphi-dation by some core reaction and regeneration bysome gram reaction.

AcknowledgmentsThis investigation was supported by the Direc-

torate-General XII of the European Communities,TNO-JET and NOES B.V. of the Netherlands.The authors acknowledge H.A. Akse, D.W.F. Brilman,J. de Haan, J. Klaus, H. Nijmeger and A.G. Steghuisfor their assistance in the experimental work andthe limestone supply companies Rheinische Kalk-steinwerke GmbH (Wtilfrath, Germany) and LhoistGroup (Otti~ies-Louva~-la-Neuve, Belgium) fortheir cooperation.

References1 S. Yagi and D. Kunii, Proc. 5th Symp. (i%t .) on Combustion195.5, Reinhold, New York, p. 231.2 J. Szekely and J.W. Evans, Chem. Eng. Sci., 25 (1970)1091.3 M. Hartman and W.R. Cough&, AIChE J., 22 (1976) 490.4 C. Georgakis, C.W. Chang and J. Szekely, Chem. Eng. Sci.,34 (7979) 1072.5 J. Szekely and M. Propster, Chem. Eng. Sci., 30 (1975)1049.6 K. DamJohansen, P.F.B. Hansen and K. lilstergaard, Chem.Erg. Sci., 46 (1991) 847.7 P.V. Ranade and D.P. Harrison, Chem. Eng. Sci., 36 (1981)1079,


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1011121314151617181920

21222324

252627

E.A. Efthimiadis and S.V. Sotirchos, Chem. Eng . Sc i . , 48(1993) 1201.PA. Ramachandran and J .M. Smith, AlChE J ., 23 (1977)353.G.R. Gavalas, AIChE J ., 26 (1980) 577.S.K. Bhatia and D.D. Perhnutter, AZChE J ., 26 (1980) 379.S.K. Bhatia, AIChE J ., 31 (1985) 642.S.E. Zarkanitis, EA. Efthimiadis and S.V. Sotirchos, Chem.Eng . Sc i . , 45 (1990) 2761.S. Reyes and K.F. J ensen, Chem. Eng . Sc i . , 42 (1987) 565.S.V. Sotirchos and S.E. Zarkanitis, AZChE J ., 3.5 (1989)1137.J .Y. Park and 0. Levenspiel, Chem. Eng . Sc i . , 30 (1975)1207.H.P. Tseng and T.F. Edgar, Fuel, 68 (1989) 114.R.H. Borgwardt and R.D. Harvey, Env i r on . Sc i . Technd . ,6 (1972) 350.L.K. Frevel and L.J . Kressley,And Chem. , 35 (1963) 1492.B. Kamphuis and U. Spitsbergen, in P.F. Sens and J .K.Wilkinson (eds.), Flue Gas and Fly Ash , Elsevier AppliedScience, London, 1989.D.W. Marsh and D.L. Ulrichson, Chem. Eng . Sc i . , 40 (1985)423.R.H. Borgwardt and K.R. Bruce, AICh.E J ., 32 (1984) 239.R.H. Borgwardt, Env i ron . Sc i . Technd . , 4 (1970) 59.R.H. Borgwardt, K.R. Bruce and J . Blake, Proc. 1st J o i n tSymp. a Dr y SO, and Sim u l tan eous SO, /NO, Con t r o lTechn i ques, San Di ego, CA, USA, 1984.R.H. Borgwardt, N. Roache and K.R. Bruce, Env i r on . P r og . ,3 (1984) 129.C.C. Sun, E.P. ONeill and D.L. Keairns, The rmoch im . Ac t a ,26 (1978) 283.C.S. Huang, F.H. Rogan and L. Kun, AZChE Symp. Ser.,77 (1981) 1.

Appendix A: NomenclatureA specific surface area (m kg- )CC concentration at core surface (mol me3)ci3 concentration at grain surface (mol mT3)

DSF

kBKKC43KPNNOP

Ir;:4RORPVPV=Jh PdVsol, eal2XX max

product layer diffusivity (m s-l OT m5mol- s-)ratio of grain radius and pore radius (pore-to-sphere factor)reaction rate constant of core reaction (ms-l CYTm4 mol- s-l)reaction rate constant of grain reaction (ms- or m4 mol- s-l)expansion factoroverall reaction rate constant core reactionlimitation) (s-l)overall reaction rate constant (grain re-action limitation) (s- )overall reaction rate constant (product-layer diffusion limitation (s- )number of pore and grain size classesinitial concentration of solid reactant ingrains (mol m -4initial weight fraction of solid reactant ingrain (purity)radius (m)radius of unreacted core (m)radius of (partly converted) grain (m)initial radius of grain (m)pore radius (m)specific pore volume (m3 kg- )molar volume of solid product (m3 mol- )molar volume of solid reactant (m3 mol- )conversion of a grain (or particle)maximum attainable conversion of a grain(or particle)

Greek lettersVi fraction of grains with initial radius R,,iPd . reac density of solid reactant (kg rnm3)+ ratio of K c and Kp

hee sink 94 grain

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