Some observations on gas-solid noncatalytic reactions with structural changes

Some observations on gas-solid noncatalytic reactions with structural changes (Received 21 July 1980; accepted 22 January 1981) Many gas-solid noncatalytic reactions such as sulfation or calcination of limestone, partial gasilication of coal, etc. are accompanied by a change in solid structure during reaction. This change in structnre affects the transport parameters within the solid pellet and hence ultimately influences the overall rate of reaction. A variable size grain model is used here to describe this situation and each grain is assumed to react according to the shrinking core model. This is a generalization of the grain model of Szekety et al.[l,Z] in that it is assumed that grains shrink or swell with reaction in order to accommodate the decreased or increased solid volume. Such an approach has also been used by Ramachandran and Smithp] in an analysis of the combined effect of tbe change in solid volume due to reaction and sintering and by Georgakis et al. in interpreting limestone suUation[4]. In parallel with this approach a closing pore model has been used by several investigators[S, 61. VANABLE SIZE GRAIN MODEL

For a gas-solid reaction of stoichiometry

The system of coupled partial differential equations (l)-(7) is solved by orthogonal collocation on finite elements[‘l, 81. Results are presented for spherical pellets and grains (F, = F, = 3). One of the key panuneters is o, the net increase (decrease) in solid volume at complete conversion per unit volume of initial voids defined by hl=(7--L)2.

When 1u< 0 one deals with partial gasifications, the pore structure becomes more open and porosity increases with the progress of reaction. When w >O porosity decreases with the progress of reaction and one can distinguiab between two cases. In Case I, as referred to in this study, comptete pore blockage never takes place since the original porosity is sufficiently large to accommodate the swelling of the solii due to reaction. Time for complete reaction of the pellet is given by:

+ -1 [ l-71 (11)

given by eqn (I)

oA(g) + bE(s) = gG(g) + sS(s).

(10)

Bl.~+l+2(;==2)

(I)

1+-.

In Case II, l o
The variable size grain model can be presented in dimensionless form by the following set of equations:

-=l-++U;:2) ‘-= I’+ F(V) dq x c f=l;

y=l B=O;

(4b) 9=1

(Sa)

where

1 (12)

l_rs~+(7+(l-7)W=I l-r

1/q i='+(&co;,-r) 1.

(13)

[

F(v) =

After time d the whole pellet reacts according to the shrinking core model which can he represented by eqns (14) and (15) below:

2-F, &

rtlP’-l[r+(l--

(6)

All the variables and parameters are defined in the Notation. The above model is based on assumptions of an isothermal pellet with initially uniform porosity and uniform size grains. No external mass transfer resistance and first order reaction with respect to gas reactant is assumed. Local effective diffusivities are taken to be proportional to local pellet porosity. The variables of interest are solid Eonversion as function of time and rate of conversion as a function of unreacted solid. They are obtained from the two equations below:

dc *P’(dc)-&ab=$&&j

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c=I.

- c

(14) (15)

Conversion is obtained from: x=1-cb+F,

I

a= 2”-‘(I-

(q(6 f))“*)d5.

(16)

RMlLTg ANDDISCUSION above can be used to study the effect of various parameters on solid convcra&t.ime behavior. At the same time the model can be incorporated into parameter estimation routines in order to evaluate the parameters from experimental conversion-time data for gas-solid non-catalytic reactions. The model formulated

(9)

2-F,

, c+ [yp+ ( I - yp)c Pr]n-Qrq

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Shorter Communications

Only a few interesting model predictions are illustrated here. In Pi. 1, the rate of conversion is plotted against the fraction of unreacted solid. For gasifications, where initially large diffusional resistances in the pellet exist (& t 1) and where a good fraction of the solid is gasified (o 4 0), the rate goes through a maximum at an intermediate conversion. This can be readily understood if one recalls that the pellet modulus is based on the initial dflusivity. If a modulus based on actual diiusivity at the exterior surface is considered, &/d/[s(l, 6)]. its value decreases rapidly as the pore structure opens up. In gasifications that have considerable diffusional limitations at the onset sufficient additional area becomes readily accessible to the gas to make the overall rate go through a maximum at intermediate conversion. At high solid conversion gasification rates approach the solution for the reaction controlled shrinking core model (for the grains) and display an apparent reaction order with respect to solid reactant of 0.67 which is to be expected for a spherical grain. For o > 0 the slowdown in the rate is proportional to the value of w. Figure 2 presents conversion-time behavior for pellets wbere complete pore closure occurs. Two different time scales are necessary for the two stages of the process one before and one after pore closure. The ratio of the grain to Pellet radius is taken to be lO_’ for all the presented results which means that stage I of the process ends at lo-* or less in time units appropriate for stage 2. This point is hence taken as zero on the time scale for stage 2. The larger the value of 01 the less conversion can be achieved in stage 1. This is the situation to be avoided in SO* capture by dolomite or limestone since residence times for the stone in reacting environment can only be of the order of the time scale for stage I. No conversion in stage 2 of the process can be practically realized due to excessive time required. Acknowledgement-This work was partially NSF Grant ENG77-23928.

supported

by the

OSCAR GARZA-GARZAt MILORAD P. DUDUKOVIt‘* Chemical Reaction Engineetiag Labomtory Department of Chemical Engineering Washington University, St. Louis MO 63136. U.S.A.

*Author to whom correspondence should be addressed. tPresent address: HYL Technologla Siderurgica, Monterray, N.L., Mexico.

aa

Ire@)

OS

oimanstmlessnm,

e

0.a

0.)

0.1,

nmn. eQ Fig. 2. Fraction of unreacted solid as a function of time in case of pore blockage (Case II). chnensionlsas

NOTATION stoichiometric coefficients for gas and solid reactants and products external surface of a solid particle dimensionless position within the pellet separating in stage 2 of reaction the partially reacted core and completely reacted solid layer, RJR,,, molar gas reactant concentration in the pellet molar concentration of gas reactant in the bulk molar solid reactant concentration in the pellet molar concentration of solid reactant in grains molar concentration of solid product in the grains effective diffusivity at a specified pellet location and time initial effective diffusivity in the pellet diffusivity through the solid product layer on the grains grain Darhkohler number, k,rJD, radio of diffusional and kinetic resistance on the grain shape factor, F = 1, 2, 3 for iminite slab, infinite cylinder and sphere, respectively modified Bessel function of the first kid of order p reaction rate constant for the unreacted shrinking core reaction of the grain position coordinate for the grain position of the shrinking core in a grain initial grain radius (F,VJA,)o position coordinate for the pellet position of the shrinking core in the pellet initial pellet radius, (F, VJA,) time volume of a solid particle solid conversion dimensionless gas concentration, CA/C, initial profile for dimensionless gas concentration dimensionless solid reactant concentration in the pellet Greek symbols volume of solid formed oer unit volume of solid reactant reacted, C&C& expansion factor for the pellet, y(1 -co) ratio of local and initial effective diffusivity in the pellet, D_J& ratio of diffusivity in the grain and initial effective diffusivity in the pellet, DiDa ratio of effective diffusivity’at the surface and initial effective diffusivity initial pellet porosity critical porosity dimensionless position of the shrinking cnre in a grain,

Lu)I

Lw

0.10

Fraim

M l_hraaGled Solid,

I-X

Fig. 1. Reaction rate as a function of fraction of umeacted solid.

rJrco dimensionless position of the shrinking core of the grains at the pellet surface at the moment of pore blockage defined by eqn (13)

Shorter Communications 8 dimensionless time based on characteristic reaction time on the grains, (b/u)(kJr&o(dC& f?’ dimensionless time at complete conversion of the pellet surfacedefinedby eqn (11) ti dimensionless time at the moment of pore blockage definedby eqn (12) & dimensionlesstime for the second stage of reaction based on c$uacteristic diltusiontime for the pellet, (@/;MW, WGJGJd(l - Eo)or &/W%%‘~J~ -

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REFERENCES [I] Szekely J. and Evans J. W., Calm Engag Sci. 197126 1901. [Z] Szekely J., Evans J. W. and Sohn H. Y., Gas So/id Reuctions, p. 160.Academic Press, New York 1976. [3] Bamacbandran P. A. and Smith J. hf., C/MI. Eagng J. 1977 14 137. [41 ~eo&ti;i~ Chang C. W. and SzekelyJ., C’hcm.EngngSci. [SI Bamachandran P. A. and Smith I. hf., A.1.Ch.E.L

1977 23

353.

6 dimensionlesspositionin the pellet, R/RP tiP

Thiele

modulusfor the pellet,

R,, J(”

5 2)

[6]

or

Chrostowski J. W. and Georgakis C., ASCSymp.

Ser. 1978 65

224. f71 Garza-Garza

O., MS Thesis. Washington University, St. Louis. Missouri, May 1979. ratio d di?ffusional [S] Garza-Garza 0. and Dudukovic M. P., Camp. Chem. Engag and kinetic resistance for the pellet 1981(in print). (u increase (decrease) in solid volume at complete conversion per unit void volume defined by eqn (10)

(R&&(Fx(l

-

q,)IkJ&/D,),

Bubble eruption diameter in a lhidized bed of large particles at elevated temperahwes (Received 3 September

1980:accepted

Large-particle tiuidized beds have become a topic of increasing interest as a result of the intensive development of fluidfxadbed coal combustors. These combustors typically use limestone particles, which have mean diameters greater than lSOO~m, to adsorb sulfur dioxide generated in the burning of high sulfur content coal. The bubble size in Ruidixed beds has received considerable attention. However, met experimental research has been done using small particles d,
8 January

I_

1981)

clearly on the film. This procedure has been used by others: e.g. 141. The average bubble eruption diameter, &, as a function of excess gas velocity, (U - U,,,,),and bed temperature is shown in Fig. 1. A combination of visual observation of the bed surface and measurement of the pressure drop across the bed was used to find ll, Fiie 2 shows good agreement between the experimentally determined values of CJ,, and those obtained from the correlation given by Wen and Yu[5]. DIXuspIoN Figure 1 indicates that the bubble eruption diameter increases significantlyas the bed temperature is increased. This is opposite the trend obtained by the investigators cited above[l,Z] who used much smaller particles and somewhat lower bed temperatures. All of the de vs (U- V,,) curves, shown in Fii. 1, have slightly positive curvature at all bed temperatures. As shown by 3c

I

I

1 nl030K /

/

Fii 1. Bubble eruption diameter at the surface of a high temperature fluidii bed, (static bed height = 0.51m).