An equilibrium—interface model for solid decomposition

AN

EQUILIBRIUM-INTERFACE MODEL DECOMPOSITION JACOB

Department of

Chemlcdl

MlJ

Englneerlng

and

D

D

SOLID

PERLMUTTER

Unlverslty of Pennsylvama

(Recewed

FOR

15 September

Phlladelphla

PA 19104

U SA

1979)

Abstract-A model IS proposed to predict the transient reactlon temperature and extent of reactlon for a sohd decomposltlon m terms ofphyslcal transport coefficients and thermodymamlcdata. allowmg also tor the effect of changing particle size arIsIng from possible density differences between reactant and product NumencaJ calculations based on the derived model detail the influences of the transport coefftcients, the heat of reactlon, and the change of particle size The results show many of the characteristics of thermal decomposltlons reported m the hterature, mcludmg s~gmo~dal conversion-time behavtor, pseudo-steadystate temperature plateaus, and self coohng Earlier models are shown to be subsumed as u-nportant special cases Because It reqmres only thermodynamic and physical transport data, the model 1sentirely Independent of any empmcal fits to kmetlc data, It 1s most applicable to high temperature, endothermic systems that produce product layers restrlctmg heat and mass transfer INTRODUCTION

are of prime concern m a great variety of chemical and metallurgical operations, including for example such diverse applications as calcmatlon [ 1,2], solid-propellant combustion [3,4], and preparation of high surfacearea metal oxides They have also received particular attention recently m connection with thermochemical cycles proposed for water-sphttmg [7, S] The models developed to describe this class of reactions &ffer greatly m their assumptions and details, dependmg m large measure on whether they emphasize chemical mechanisms or the associated physical processes Those that focus on chemical steps are especially hard to generahze, but attempts m this dlrectlon may be found m Garner [9] and Young [lo] Such models have commonly been used to analyze experunental data or simulate decomposltlon rates isothermal or under either non-Isothermal condltlons [l l-141, but Johnson and Gallagher [ 1.51, Satava [ 161, and Wendlandt [17] have pointed out that such determmatlons of the rate-controlled process are ambiguous owing to the small differences among the predlctlons of the individual model functions The assumption of an Isothermal particle IS m any event questionable, because solid decomposltlon reactmns are activated and endothermic A temperature gradlent between the furnace atmosphere and the reactant IS necessary to ensure the heat supply that sustams the reactlons Satterfield and Feakes [lg, 191 measured true reactlon temperatures m decomposmg pellets They discovered that these temperatures remained effectively constant over wide conversion ranges and reported differences of up to 140°C between the reacting pellet and the furnace This temperature plateau was independently observed by Hllls[20] Draper and Sveum [21] indicated that the decomposItIon reaction of a sohd IS controlled prlmarlly by Sohd

decomposltlon

reactlons

the heat transfer process Gallagher and Johnson [22] also mentioned the possible effect of self-coohng behavior Ruckenstem and Vavanellos [23,24] dealt wtth the kinetics based on nucleation theory by taking mto account heat losses through convection and radiation from a reacting solid and showed the influence of the heat transfer parameters m mterpretmg experunental observations DIffusional effects m gas-sohd reactions have been accounted for m many model developments [25-291 If the Internal dll~usronal re\lstance I\ ncgllglble mass transfer resistance m the gas-sohd boundary layer starts to play a role The extent of its Influence depends on the external condltlon m the flowing gas and the sample size Shen and Smith [26], and Rehmot and Saxena [30] took mto account the changes of particle size that arise m a gas-sohd reaction when the densities of solid reactants and products differ Hills [20] derived models for decomposltlon reaction which included detalled consrderatlon of heat and mass

transfer

effects MODLL

DEV

FLOPMENT

For a sohd decomposltlon that produces a gas product according to the reaction

S(s) -WA(g)

+ bB(s)

the well-known shrinking core model [25,31] gives the gas product profile m the reactant layer of a spherical particle as

Y

=

x* +

o!(rc)

; -

:

1

[

where

x* -

WC) = 5

D

kc+-,’

X0

c-3

_‘+’ c

c

JACOBMu and D D PERLMU~ER

1646

and the temperature at the surface of the reactive core as 0, = -&m(c)

1

1 k -+y-hRr, r,

r,

1

+ 1

(3)

Knudsen type, for example, DC IS proportional to the square root of absolute temperature, for more ordinary molecular dlffuslon, the exponent on the temperature ratio IS m the range of 1 5 to 2 0 [33] To generalize D,=Dy

where

DC =

(9)

and

AHC;: DC

(10)

B,=PC

akT 0

The assumptions nnpliclt in this set of equations may be listed as follows (1) Depending on the density of the solid product m comparison to the reactant, the particle will either grow or shrmk as the reaction proceeds, but m any case there will be a well-defined boundary between the product layer and an unreacted core Shen and Smith [26] and Rehmot and Saxena [30] have shown how this change may be accounted for m terms of the ratio of the relative volumes of the two solids

where 0 5 I m 5 2 To depart from the dn-ectlon of prior developments, consider that it IS not necessary to treat the rate of reaction as controlled by a postulated surface kmetlcs, especially for a solid decomposltlon reaction where an order of reaction IS not a well-defined concept with respect to solid concentration Assummg instead that the reaction at the surface of the unreacted core IS fast enough to mamtam the eqmhbnum concentration

(11) (2) The gas product diffuses through the solid product layer to the surface of the pellet Heat must be transferred through both the gas boundary layer and the solid product layer to the surface of the inner core to sustam the endothermic reaction Blschoff [32] has shown that for such moving boundary dlffuslon problems the pseudo-steady-state-hypothesis (PSSH) ISvalid for solid-gas systems, that IS,it ISan acceptable approxlmatlon to consider dtffuslve changes as occurrmg rapidly m comparison with the relatively slower rate of movement of the interface mass transfer (3) Conventional heat and coefficients determme the usual boundary condltlons at the particle surface dx dr-_

_-

d8 dr

k,R D, hR =k(8-

(x0 -

xl

The rate of shrinkage of the inner core IS m &mensionless form

dr,_--z dr

WWr, 1 r,

(12)

With mltlal condltlon r, = 1 at T = 0, integration of eqn (12) sves 2=6

(13)

which by substltutlon of eqn (2) becomes

atr = rs 7=6

1)

(7) (14)

(4) The endothermic heat of reaction IS supplied at the core surface solely by the flux of new product leavmg this boundary Formally

where the heat and mass tral sfer coefficients have been wrltten as functlonq of particle size followmlr the wellknown correlations of Rowe et al [34] and Rehmat and Saxena [30]

(8) It should be noted that & IS not a constant but a function of temperature as a consequence of the dependence of dlffuslvlty on temperature The specific form depends on the diffusional mechanism that IS applicable to the case at hand If the dlffuslon IS of the

k,=%[$+-+]

(15)

and

h=;[F++]

(16)

NOISt13AN03

‘X

.S ----

------

-----

. 0

ii

NOlSU3AN03

‘X

Ir --

---*

v)

ii

3WlLVk43dW31

NOIL3V38

‘=%

An eqmhbnun-Interface

model for sohd decomposltlon

p, DIMENSIONLESS Fig

3

The effect of p on temperature plateau at different values of (AH/ai%JeT,) z = 1, m = 0 5, K1 = 10, K, =5 K, =4, K, =2

1649

JACOB Muand

1650

D

D

PERLMU-ITER

[AH/a

6LT,l -10

I AH/a blT,) =20

0

05

10

15

T, DIMENSIONLESS

20 REACTION

25

35

30

TIME

35

I0

I

: : 1’ [AH/a&T,1 (AH/a

=30

&To1 -20

I AH/a CXT,) = 10

Fag 4 Effect of endothermic heat of reactlon on conversion and reactlon temperature development m = 0 5, K, =lO

K,=5,K,=4

K,=2,r=l

An equdlbnun-Interface.

0

3

1651

model for sohd decomposxtlon

6 T, DIMENS1ONLESS

9 REACTION

12

15

TIME

5 Effect of @ on conversmn and reaction temperature development m = 0 5, K1 = 10, K2 = 5, K3 = 4, K., = 2, z = 1, (AHfa.9?To) = 20

JACOB Mu and D D PERLMUTTER

1652

T, DIMENSIONLESS

REACTION

TIME

0 97

096

[

Fig 6 Effectofdlfferentcombmatlonsofboundarylayerheat and mass transfer coefficients on conversIon and reaction temperature development m = 1 5, z = 1, p = 005, (AH/aSeTO)= 20

they&d not speczfically mention the shape. of the curve they expected The most unportant feature of the model presented here ISits independence ofemplrlcal curve fitting, usmg only thermodynamic data (heat of reactlon) and physical data (transport coefficients) to predict extent of conversion and reaction temperature for a solid decomposltlon To compare the computed results of this study with the predlctlons of competmg models, three computed solutions are presented in Fig 7 for typlcal values of the several system parameters Curves A and B are from the model developed here, dlffermg only with respect to the boundary layer transfer coefficients the former IS for moderate levels of these parameters, and the latter IS based on the supposmon of relatively large heat and mass transfer coefficients Curve C on Fig 7 IS obtamed from Yagl and Kunu’s [25] predlctlon for the special case of an isothermal particle As antlclpated by the defimtlon of the dlmenslonless time scale, the reactlon ~111 be complete at T = 1, if the reactant IS heated to the

external gas temperature very rapidly This will not be achieved for practical transport rates, however, as shown by curve A for which complete reactlon reqmres ,c>2 The curve B ISincluded m Fig 7 to demonstrate that even If the boundary layer heat and mass transfer coefficients become very large, the temperature gradient still persists wlthm the particle and the conversion curve IS posItIoned between curves A and C It should be noted that the Isothermal approxlmatlon can produce appreciable errors m the prediction of converslon times, even in cases where the particle reaction temperature levels off within 3 % of 0, = 1 Smce a considerable prior hterature exists on the calcmatlon reactlon of calcmm carbonate [ l&20,22], this system provides a good basis for comparmg model predlctlons with expertmental data Moreover, because the calcmatlon reactlon (I) 1s highly endothermtc, (II) produces a product layer that severely hmlts heat and mass transfer, and (111)takes place at

An eqiuhbnun-Interface

I

I

05

I

10 ‘c, OlMENSlONLESS

Fig

7

1653

model for sohd decomposltlon

t

15 REACTION

I

I

20

25

TIME

A comparison of three model predtctlons wtth respect to conversion and reactlon temperature development m = 0 5, = = 10 K, =10,K,=5,K,=4,K,=2,(AH/a9?T0)=20,/?=OOS

relatively high temperatures, the assumptions of this model are hkely to be Justified for this sohd decomposltlon Hills [20] measured the particle reaction temperature for this system with an embedded thermocouple, and reaffirmed the initial transient temperature characterstics found by Satterfield and Feakes [ 181 Hls’ expernnental data are presented m Fig. 8 together with the pre&ctlons of the several models to be compared Curve A IS the result of the model developed here, computed by usmg the expervnental con&tlons and relevant physical and thermal properties The delis of calculation are ltemlzed m Table 1 Curve B IS the predicted result of applymg the product layer Muslon mode1 with temperature corretion as formulated by eqns (17) and (18), and curve C corresponds to the isothermal product layer diffusion model of Yag! and Kunn [25]

Above all, these comparisons show the effect of thermal penetration mto the particle If isothermal conditions are established mstantaneously, curve C shows the rapid completion of reaction at T = 1 If the transfer coefiiclents are taken to be very large m comparison with mtrapartlcle transport, the rate of reaction IS slowed somewhat as shown 111curve B, and prtictlon 1s mtermedlate between the isothermal case and the more reahstlc curve A The temperatures measured by Hills are shown m Fig.9 where two mode1 pre&ctlons are also supernnposed The sohd curve 1s that pre&cted by the equations developed above. and the dashed hne 1sfrom H11ls’model [20] The new result I\ cvldentlv clo\er to the data, especially 111the mitial stages of reaction In this connectfon it-should by noted that the dlstmctlon between these models only shows up m the

1654

JACOB

Mu

and D D PERLMUTTER

096

092

0 90 0

05

10 'T, DIMENSIONLESS

15 REACTION

TIME

20

25

FIN 8 A comparison of three model predIctIons and data from Hills [20] on the thermal decomposmon

of

calcium carbonate Table 1 Detads of parameters used to calculateconversions and reaction temperatures

Variable Reynolds

number

Symbol

Result (drmenslonleqs)

4e

0 62*

Schmidt

number

SC

1 27

Prandtl

number

P??

0 74

m

0

5

size factor from Equatxon (5)

1

11

Mass

transfer

constant

5 95

Mass

transfer

constant

1

75

Heat

transfer

constant

0

27

Heat

transfer

constant

0

07

Dlffuslon temperature factor

CoefflClent correction

aWaRT Dimensionless rat10 Of the heat of reactlon to the effective heat conductlvlty of the solId product layer

*Corrected from Hdls

17

79

0 0257

[20] report by usmg actual gas density at 902’C

temperature predlctlons, but that the conversion curves for both models are mdlstmgmshable on the coordmates of Fig 8 This may be the result of a fortuitous compensation effect, smce Hills’ equation does not take mto account either the gas phase concentration of the product or the particle size change Calculations based on eqn (5) mdlcate that z = 1 1 and estnnates of x0 indicate that It may be as large as 0 5 m the untml stages of reactlon To augment the temperature measurements taken from the literature, mdependent measurements were run m our own laboratory on the kinetics of calclum

carbonate calcmatlon The sample was of reagent grade (Fisher Sclenttic) with purity of at least 99 3 wt %, particle size of 20 to 60 ,um, and density 2 7 1 g/cm3 To conform with the requirements of the model presented her, the solid reactant must have a well-defined boundary surface. The powder was therefore compressed m a pelletizer (Arthur B Thomas model 7880-F15) mto cyhndncal pellets of 2mm dla and 2 2 mm length, to provide a pellet density of 2 12 g/cm3 In view of the virtual lmposslblllty of conducting truly isothermal reactlons on endotbernuc sohd

An eqmllbrmm-Interfacemodel for solid decomposltlon

1655

r

940

7LO 1

0

I

I

05

I

10 T,

I

I5

DlMENSlONLESS

I

20

REACTION

25

TIME

Fig 9 A comparisonof two model predlctlonsand data from Hills [20] (I 9681on thethermaldecomposmon of calclum carbonate

02

0

500

600 T,,,

Fig

10 Two mod4

predlctlons

700 FURNACE

TEMPERATURE,

800 “C

and actual data for the thermal decomposmon thermograwmetrlc analyzer

decomposltlons, a temperature-pregrammed thermogravnnetrlc analyzer (DuPont model 951 TGA) was used to study the kmetuzs at a slow heatmg rate of lO”C/mm To dnve off the gas product of reactlon, a nitrogen atmosphere was mamtamed over the sample by d flow of 80 cm3/mm through the 2 5 cm dla furnace tube The reaction was first detected at about 560°C and completed by 800°C The test of Fig 10 demonstrates that the prehctlve curve follows the we&t loss data closely, considerably nnproved over the former Yagl and Kunu model, even though the cyhndrlcal pellets only approximated the spherlcal shape called for m the model With regard to such TGA results, It IS of primary nnportance that the model developed here IS

of CaCO,

m a

apphcable not only to lsothezmal studies, but especially when furnace temperature varies m a known manner The equations do account for changes m T,, provtded they are relatively slow research was supportedby the U S Departmentof Energy,office of BasicEnergy Sciencesunder contract No EY-76-S-02-2747

Acknowledgement-Thrs

NOTATION

a

b Cz D

stolchlometnc coefficient stolchlometnc coefficient gas product concentration m the bulk gas phase m eqmhbnum with solid at temperature T, effect dfluslon coefficient of gas product A through solid product layer at temperature T,

1656

0‘

h hc h,

AH

k

JACOB MU and D D PFRLMIJTTFR

dlffuslon coefficient ofgas product A m the bulk gas phase effective diffusion coefficient of gas product A through the solid product layer at position r = r, effective dlffuslon coefficient of gas product A through the solid product layer at position r = r, overall heat transfer coefficient at r = rs convective heat transfer coefficient at r = ‘; radiation heat transfer coefficient at r = r, heat of decomposltlon reactlon effective thermal conductlvlty of the solid product layer thermal conductlvlty of the bulk gas phase mass transfer coefficient of gas product A Do/D, 0 35(0,/D,)

(SC)“’

(Re)“’

kolk(l + h,lh,) 0 35(k,/k) (Pr)‘13 (Re)“’ (1 + /q/h,) dlffuslon coefficient temperature correction factor M molecular weight of the solid reactant S molecular weight of the solid product B MB Pr Prandtl number r dlmenslonless radial distance from the center of the pellet ‘E dimensionless radius of the unreacted core dlmenslonless radius of the pellet ii initial particle size $39 gas constant Re Reynolds number SC Schmidt number t reaction time temperature of the bulk gas phase T, X dlmenslonless concentration of gas product A dunenslonless concentration of gas product A m X0 the bulk gas phase X* dlmenslonless concentration of gas product A at position rc X conversion = 1 - r,” z sn.e factor defined m eqn (5) Greek

symbols

defined m eqn (2) ,6 value of #i, at temperature T, j?, defined m eqn (4) 8 dlmenslonless temperature tic dimensionless temperature at position r, T dlmenslonless reaction time = DC: Mt/6apK* p density of sohd reactant S ps density of sohd product B

a(rL)

REFERENCES [I

Coats A

W

J P

R , Hills A W

D

Nature

1964

201 68

Chcm EnqrUz % r 1970 25 929 [3 Klshore K, and Verneker V R P AIAA 1976 14 966 [4 Herley P J, Reactzmty of Soltds Chapman & Hall, London 1972 [5 Fox P G , Ehretsmann J and Brown C E , J Catal 1970 20 67 [6] Dollunore D and NIcholson D , J Chem Sot 1962 906 processes for hydrogen [7;Cox K E, Thermochemrcal production Los Alamos Screntlfic Laboratory Report LA-6970-PR 1977 [S] Krikorlan 0 H , Pearson R K Otsukl H H , and Elson R E , Chemical and process design studres ofthermochemrcal cycles for hydrogen productzon Lawrence LIvermore Laboratory Report W-7405-Eng-48 1977 [91 Garner W C Chemzstry ofthe Sold State Butterworths, London 1955 [lo] Young D A Drcumpovtton of Jolrds Pergamon Press, Oxford 1966 [ll Davies J V, Jacobs P W M and Russell-Jones A, Trans Faraday Sot 1967 63(7) 1737 [123 SzunlewlczR and MantltlusA, Chrm tngng J~I 1974 29 1701 Cl31 Verneker V R Pal Thermorhzm Acta 1975 13 293 r143 Bcckm inn J W Tlttrmr~ch~~~ Icrrc 1977 19 I IX [15] Johnson D W, Jr and Gallagher P K ,J Am Cer~tm Sot 1971 54 461 [16 Satava, V , Thermochrm Acta 1971 2 423 [I7 Wendlendt W W, Thermochrm Acta 1973 7 476 [18] Satterheld C N and Feakes F, A 1 Ch E. J 1958 5 115 El91 Satterfield C N and Feakes F , A I Ch E J 1959 6 122 [2U] Hdls A W D C/lam Engng Scr 1968 23 297 [21] Draper A L and Sveum L K , Thrrmochzm Actu 1970 1 345 [22] Gallagher P K and Johnson D W , Thermorhrm Acre 1977 6 67 123 Kuckenstem E and Vavanellos T, A I Ch E J 1975 21 756 [24 Ruckenstem E and VavanelIos T , A I Ch E J 1976 22 376 [25 Yagl and Kunu, 5th Symp Int Combustwn Remhold, New York 1955 [26] Shen T and Smith J M , Ind Eugrtg C hem b undb 1965 4 293 r27’lshldaM andWenC Y AIChEJ 196814311 [28] Szekely J , dnd Propster M Chem Engng Scr 1975 30 1049 [29] Park J Y and Levensplel 0 , Chem Engng Scr 1975 30 1207 [30] RehmatA andSaxenaS C , Ind Engng Chrm Proc Dev Dev 1976 15 343 [311 Levensplel 0, Chemxnl Reacrwn Engzneermg Wiley, New York 1962 [32 Blschoff K B, Chem Engng SCI 1963 18 711 [33] Bird R B , Stewart W E and LIghtfoot E N , Transporr Phennmenn Wiley New York 1960 [34 Rowe R N , Claxton K T and Lewis J B, Trnns lnstn Chem Engrs 1965 43 14 [35_ Rozovskl A Y, Kmetrka I Katnlrz 1962 56 780 136 Boldyrev V V, J Thermal AnaI 1975 7 685 [37 Boldyrev V V, J Thermal Anal 1975 S(1) 175 [Z’ Campell

F

and Redfern

and Pauhn A