MODELLING OF FLUIDIZED BED REACTORS-V

COMBUSTION OF CARBON PARTICLES-AN EXTENSION

ALFRED0 L GORDCYN,t HUGO S CARAM# and NEAL R AMUNDSONB Department of Chemical Engmeermg and Mater& Sctence, Umverslty of Mmnesota, Mmneapolts, MN 55455,

USA

(Recewed 6 June 1977. received for prrbllcafzon 6 October 1977)

Absbmet-The problem of the bur&ng of carbon particles in a fluId& bed m which oxygen and carbon dtoxlde react at the surface of the park&s and the carbon monolude produced bums m both the dilute phase and mterstlhal gas of the dense phase usmg the DavIdson-HarrIson model IS solved The numerkal solution of thrs problem IS severe Results of the general model are compared with two sunpler models Parametnc studies are made for dtierent pamele SIZS, mlet temperatures, flow velocltles. and bubble duuneters

INTRoDucTloN

In our recent paper111 a model for contmuous com- bustion of carbon par&les m a non-Isothermal fluuhzed bed was developed based on the two phase theoryi for the dlvrslon of gas flow There it was consldered that the pmcles were burned by two heterogeneous reactions oxldatron by oxygen and reduction of carbon &oxlde Both reactions produce carbon monoxide whrch IS ox+ dued by mcommg au The complete set of mass and heat conservation equations were solved for the case m which no oxidation of carbon monoxide occurred withm the bubbles and when the rate of the heterogeneous reac- tlons at the surface of the particles was much faster than the rate of dlffuston of reactants However, some num- mencal results[3], m parkular for the case in wluch the fraction of dilute phase tn the bed IS high, mdlcate that the homogeneous reaction wIthin the bubbles may be signdicant and can not be neglected as u1 Ref 111 On the other hand, it was suggested that the assumptron of mass transfer control m&t not always be valid The purpose of this paper IS to treat the general model and to compare the results with the sunpler case. Th8s represents a considerable numerical and theoretical comphcation and some of the theorehcal results obtamed m Ref [I] are not possible

TuEIbllXXL The basic assumptions are the same as m Ref [l] and

wdl not be repeated here The not&on used tn thts paper IS the same as in the previous paper unless mdlcated otherwise For the sake of clarity only the dunenslonless mass and energy conservation equations for each com- ponent and phase wdl be restated

*Department de Ingemcna Qutmlca, Casdla 53-C. Umvefsuiad de Conctpclon, CNe, South America.

Sfkpartment of Chemical Engmecnng, LehM Umverslty, Bethlehem, PA 18015. U S A

ODepartment of Chemical Engmeermg. Untverslty of Houston, Houston, TX 77004, U S A

D&e (bubble) phase

- $j+ K&J& - yJ + a&U ew (-WY~)Y,YZ = 0,

J= 1,2,3 (I)

Yi(O) = YP (la) _

- 2 + H~&(x. - YJ + &L u w (---NJYJY ,Y, = 0

(2)

Y.03 = 1 (W

Interstitral gas

yp-x,+aK~~(~~-x~)+3Wp ‘(Gs-q~(r)dr I 0

+ a&l2 exp (-NJxq)x1x2 = 0, J= 1,2,3 (3)

1-%4+aH~~(~4-~4)+3W;r~ I

’ [T,(rI - x@dr) dr 0 r’

+ H,(T,, -x4) + KzL2 exp (-NJx.,)x,xz = 0 (4)

Soirds Cfor each particle) Concentrations of reactants and products at the sur-

face

xj - C, + aIrKtrexp(-NJfp)CI. + a3&r exp (-NJT,)C,, = 0.

J = 1.2,3 (5)

Rate of shrmkage

dr R(r)=*= -2K, DA4 exp (-NJ T,)C,,

- K,DM exp (-NJ T&Z,, (61

r(O)= 1 (W

713

714

Temperature

dT,+2 T &r= 3DLIKI exp (--N,IT,)C,, de r PdB PI

A L GORDON et al

From eqn (5) we get C,. and C,, and these are substituted Into eqn (6) to get a moddied relation for R(r) With thts and with C,, and C,, Introduced m eqn (7) there results

+3DL,K, exp (-NJT,)G Pr

+v (7)

T,(O) = T,” Ua)

Performrng equation

‘r”dr r=- Rlr) (8)

while

I

1 f= o adz

The above equations represent

(9)

a model henceforth referred to as the G-Model In order to solve this system of equations, rt IS transformed as follows

Equation (5) can be rearranged to obtam

c,. -x, = *Ii KI r exp (- N, I Tp)xI I+ K,rexp (-NJT,)

K3rexp(-N3/Tp)x3 + *” 1+ Kg exp (-NJ T,) (10)

It was shown m Ref [l] that

Fo 2 (11)

and mtegratmg eqn (11) gives (rn dlmenstonless form)

(12)

Then eqn (IO) IS substituted mto eqn (3) and eqn (12) IS substituted m the resultmg relation, as well as m eqn (4) to obtam

YP-- x, + UK,&, - 4

3U,‘W,Dx, I I

K,rZexp(-N,/T,)dr - 7 0 RW[l+ K,r exp (-N,/T,)I

_ 3a3, W,fi3 I K3? exp (-NJT,) dr

7 R(r)D + K3rexp (-N31TP)I

+ a2,K2 exp (-N21x4)x,x2 = 0, I= 1.2.3

and

1 - x4 + OHI&. - xi1 - 3 w,WP ’ (T,(r) - x4)r dr 7 R(r)

+ HAT,, - x4) + K& exp (-N&Jxlxz = 0

(13)

(14)

%$=_~T~+(‘D~Ix~) K,exp(-N,IT,) R(r)r[l + K,rexp (-NJT,)]

K3 exp (-N,IT,) R(r)r[l + K3r exp (-N3/TP)]

(15)

T,(l) = T,” Wa)

Thus the problem has been reduced to the system of eqns (1). (2), (13)-(H) In short, the system can be represented by the followmg

$ = f(x, Y)

Ym = Y”

4(x, ?. Ii) = 0, I = 1,2,3,4

z=g(r, T,,x)

T,(l) = T,”

where

‘I= o R(r)[l+K,rexp(-NJT,)] I

1 K,?exp(- N,IT,)dr

I

I z* =

K32 exp (-N3/ T,) dr 0 R(r)[l + K3r exp (-N3/Tp)l

z, = I

‘(T, -xJrdr 0 R(r)

I, =

(16)

(1W

(17)

(18)

@a)

(19)

(20)

(21)

(22)

Although the existence and umqueness of a solution to this set of equations wdI not be establlshed, we will assume that an mzratlve procedure wdl produce the solution The tteratlve process couId be as follows An uultlal guess x0 IS chosen With this substituted mto eqn (16) one obtams

dy”’ dr = f(x”, Jr@-)

and by quadrature

-to> = Y

On the other hand, with x““, eqn (15) can be numerIcally Integrated to obtam T,“‘(r) m the mtezval (1,0) For the integration R(r) can be expressed as a function of x and T,, as follows,

Modelhng of fluldlzed bed reactors-V 715

R(r) = - 2K,LXU exp (-N,/T,)x, atomic spectes to be conserved whtch lmpltes that two

1 + K,r exp (-N,/T’_) Independent variables should be suf6ctent to descnbe all _ K&U exp (- NJT&, possslble changes of wmposttion of the gas phase Equ-

1 + K3r exp (-NJ T,) atlons (13) and (14) can then be explicitly wrttten

With known 7’?(r) the mtegrals gven m eqns (19)-(22) can be evaluated by quadrature and a new vector x IS obtamed from eqn (17) by means of a Newton-Raphson algonthm expressed m general as

X (*+I) = $k) _ [~(xw)]-‘~(Xw), k=O, 1,

provided the inverse exists Because of the relation

4(x, 9, I,) = 0, 1 = 1,2,3,4 (17)

it IS clear that m determuung the +I~~c?x, m the matrix 4 we will need terms of the form

3 ax, and 2

1

They can be evaluated by the mtegratton of the proper vanatlonal equations. whose dertvatton can be found m Ref [31 Although simple tn pnnctple, the method presents several pract~al dzfiicultles that are described below which result partly from the model Itself and partly from the algorithm described above and mdlcate the need for a somewhat different approach In fact, m a number of tru& the system did not converge to a solu- tion, or else required a prohrbltlvely large number of tterattons The frulure of convergence was clearly due to the use of the Newton-Raphson method which IS a local exploration and gives only the root sufficiently close to the mtttal guess In our problem a vector of four com- ponents must be guessed so the chance of maktng a poor choice IS evident It 1s useful m makmg thts guess to solve a simpler problem and use the result as an uuttal value for the general case Even domg this and often no matter how close the uuti guess was to the solution, the procedure would diverge or osctiate

On the other hand, there were numerical ddliculttes artsing from the large number of equuatrons to be solved (sutteen for aydax,, four for the average values ajqax,, four for aTdax, and sutteen for Ajax,) and from the form of the equations themselves As seen before, a number of vanattonal equattons must be solved sunul- taneously with the ongtnal set and there were ranges of vanables and parameters which made the equ.&ons very staff

It appears clear now that rt would be convenient to reduce the number of unknowns m the system and to devise a sunpler method of solutmn that would avoid the cumbersome solution of the vartational equattons and define a stmplrfied model that would provide convement guesses for the solution of the G-model

Reducrron of the number of unknowns m the model There are three gaseous components to be considered

oxygen, carbon monoxide, and carbon dloxlde and one

Y1 * - XI + aKl,,(y’, - xl) - J,x, - J3 = 0 (23)

Y,” - x, + aK&j$ - x,) + U,x, + 2J,x,- 2J, = 0 (24)

yXo - x3 + aK&j$ - x3) - 4x, + 2J, = 0 (25)

1 - x4 + aH,(jTd - x.,) - J4 + H,( Tws - x4) + L& = 0 (26)

where

, =_3WJ% I 7 ’ , __3wp1, 2- 7 ’

J3 = Kz exp (- NJx3xIxs J 4

= 3 KSYP& 7 ’ ml

and defimug Z, = J2xj - 2.J,, Z, = J,x, + J3 we can sub- stttute into the mass balance equations to obtam

y,‘- x, + aK,,(y’, - x,) - Z, = 0

y,” - x, + OK,& & - x2) + 22, + 22, = 0

Y,O- x3 + CYK,~( y3 - x,) - Z, = 0

If we now wnte eqns (1) and (2) exphctly

dy, _ z - KrJ.J(x~ - Y 0 - Js

dyz _ z - KmU(x, - ~2) - Zr,

2 = K,,U(x, - yp) + 2J,

I

(1)

and with

2 = Hd’(x, - ~4) + LA (2)

J5 = K2u em (-&/Y.)Y,Y~ (31a)

Eltmtnatmg J5 from two of the mass balance eqn (1) we obtam

$ (2~ I - ~3 = &&(2x, - x2 - (2~ I- ~3)

$ (2~ 1+ ~3) = &,U(2x, - x3 - (2~ I + ~3)

wluch can be mtegrated to yteld

y, = 2y, + [2x, - x2 - (2y,* - y2q] e--ic~Dprr* - (2x, - x2)

(32)

y3 = -2y, - [2x, + x3 - (2y,O + ya7] e-KmU* +(2x, + x3)

(33)

CES Vd 33 No 6-F

Takmg the average values along the reactor

~~=281+12x,-x2-(2y,0-Y241~-(2*,-x3 (34) ID

P = + [ 1 - exp (- UK,,)]

as before, and, If we know the condltlons m the dense phase, the problem IS reduced to the solution of the followmg dtierentlal equations

~=K,“U(X,-Y,)-K,U~~P(-N,/Y~)Y,Y, (36)

Equation (41) IS equivalent to the assumption of constant total concentration of oxygen atoms and 1s obtamed by combmmg eqns (29)-(31) to elunmate Z1 and Z, and use eqns (34) and (35) to ehmmate yz;, j3 and y’,

(2) We can now mtegrate eqns (36) and (37) to obtam y’, and Y. and with them the value Yz

(3) With the values aven m step (1). calculate the integrals defined III eqns (19)-(22)

(4) With the values obtamed m steps (2) and (3) we may solve an approximate version of eqns (23), (24) and (26) as follows

y,‘-X, + d&(y’, -Xl)- &XI -J, = 0 (a)

yz” - x2 + aK&-,(j$ - x2) + 2J,x, + J2(2yIo + yzo + 2y30 - 2x, - x3 - 2J3 = 0 (b)

Choosing JS as the unknown we find

2 = H,Du(& - Yd + L&U exP(- Nz/ydy,yz (37) x~ = h”+a&D~I - J3

I+aKID+Ja

with the mltml condlhons gven m (la) and (2.a) If the coetficlents KID and HI0 are found from the Davidson

x2 _ Yo”+ o&&+Jlxl r:fil” z F”’ 2y30 - 2xl) - u3 ID 2

and Hamson model and If we set 9 = Ai& and HID = KID then we can ehmmate one more dlfferentlal equation x4 =

1 + a&& - 14 + &T*, + ,533

l+crH,+H, obtammg

y* = -L,y, - [Lzx, + x4 - (L*y,“+ 1)] e-KlD”z + Lg, +x4 and we must solve one nor&near algebmc equation in J3

(38) J3 = Kz exp (-N21x4(J3))x,(J3)x2(J~)

and With this value of J3 we obtam agam new values of xl,

L= -L,y’,-Lx, +x4-&YI”+ 111 g+L2x,+x. x2, and x, to be used to repeat the process ID

(39) (5) The iteration process stops d a norm defined as

and the concentration dlstrrbutlon m the dilute phase can be obtamed by integration of a smgle tist order dlfferen-

l/xi = ,_J= 4 1X:“+” - X:=‘l*

tml equation 1s less than a preset value which depends on the

dY,

accuracy desired

dr = &DU(-% - Yd- K,U exP (-N~/Y&, Ydy,y, In this way the number of calculations per iteration is

notably reduced Agam, convergence depends on the (40) uutlal guess made, and that was provided by the sim-

If we have a reasonable guess of the state of the system, pIGed model which we wdl descnbe next Once one

we can now generate a somewhat sunpler method of solution IS obtamed, additional results are produced by

solution than the one proposed before and which avoids changmg gradually and shghtly the set of independent

the calculation of the varmtlonal equations As we know, parameters We note, however. that compumons are

our ownal problem was to solve stdl very tune consummg and that only a hmlted number of runs were camed out

Nx, 5, I,) = 0, I = 1,2.3,4 (17) TmESMuDEL

and usmg the previous analysis the method proposed wdl It was remarked m the Introduction that the simpltiy- be as follows mg assumptions made m Ref Cl] do not always hold In

(1) Assume values for any patr of concentrations, say parucular. it may be necessary to take into account the x, and x, and the temperature The thud concentration homogeneous reactmn between carbon monoxide and x, m thus case IS gven by oxygen wlthm the bubbles In view of the many d&z&-

ties m obtammg the solution of the general problem x =2Y,“+Y20+2Y30-(2x,+x*)

3 (41) proposed above, It seemed worthwhtie to explore pos-

2 sable solutions for a simpler model but one that retams

Modelhng of flu&zed bed reactors--V 717

homogeneous reaction m the ddute phase A snnpler model and one consldered earlier IS that m which we would have mass transfer control at the parhcle surface but wth reactlon m the drtute phase ms is a shght generahzatlon of the model consldered m detad m the previous paper We wdl refer to thts model as the Rx model Equations (l)-(8) reduce to the followmng system

- 2 + K,,U(x, - x,) + &G U exp (--NJYJY,Y~ = 0.

J = 1,2,3 (42)

Yi(O) = YP (42a)

- 2 •t H,lJ(x, - YJ+ KJJJ exp (--NJY~Y,Y~ = 0

(43)

Y.Go = 1 WI

~,‘-xi+~K,~(~--,)-f5~,,W~,+~a,W~~

+ a,& exp (-NJxJxIx2 = 0. J = 1.2.3 (45)

I- x4 + aH&j$ - x4) + 5 W,D(2pMT,’ + L,)x,

+ 5 W,D(juUTPo + L3)xJ + H,( T,, - x4)

+ K,L, exp (- NJxJxIxZ = 0 (46)

The same analysis camed out earher can be performed m this case, but the sunphfications can be taken further than m the prevrous cases and the system reduced to a system of no&near equations with two unknowns m the general case (HI& K,,,) and a smgle nonhnear equation when K,D = HID Substltutmg the values of (A- x,.) and (&-x,) obtamed from eqns (34) and (35) m the mass conservation equations for carbon monoxide and carbon dioxide and ehmmatmg the rate of combustion of carbon monoxide m the mterst&al phase between them and the conservation equation for oxygen we obtam

x3 = -(l + aP)[Zx, - (2y,O + y341- lo& w, 1 +u.P+5w,D (47)

x*= (1 + apwX, - (2Y,O- y*41+ 20 W&r + 10 WLDXX, (1 +&P)

(48)

We wdl have then to solve sunultaneously

y,“-x,+aK,&,-x,)-5W,Dx,-Js=O

1 - x, + rrw,( y4 - x4) + 5 W,IJ(2&fTPo + L ,)x1

+ 5 WfipMT,’ + Ldx, + H,( T,, - x4) + L,J, = 0

J3 = & exp (- N2/x4~x1x2

together with eqns (36) and (37) which wdl provide the values of y’, and jd If Hn, = Kin we can go a step further and using eqn (38) ehmmate & - x.1 to obtam

and we only have to solve

Y lo - x1 + dh,(y’, - x,1 - 5 W-1 - K2 exp (-N2/x4(xI))~1~2 = 0 (50)

with the value of 9, to be obtamed from the integration of eqn (40) It should be noted that we can m both cases restnct the search to the values of x, which wdl generate posltlve values of x2 and x3 m eqns (47) and (48)

NUMERICAL CALCULATIONS OF STEADY =ATEs @OR THE %-MODEL)

Presented here are the numerlcal calculation of steady- states, and the mlluence of a selected number of parameters on the mulUphclty pattern Smce the com- putation time for the Rx model is about ten tunes that for the model used in Ref [l], and much more for the G-model, we concentrated on those sets of parameters for which a useful compmson among models could be made In all numerical examples the values of the physl- cal constants used are taken from Table 5 of Ref [l], and the dunenslonless groups have been evaluated usmg the correlations even m Table 1 of that paper

lNFUJENCEOFTHEJNUTGASlIATE

The dependence of the system upon the flow rate and Its temperature can be seen in FQS 1 and 2 as well as m some values of Table 1 The curves in those Ggures are the steady state values found by varying the flow rate of gas under a fixed mfiuent temperature Table 1 should be compared with Table 6 of Ref [l] It can be observed that for the same sets of independent parameters the values of temperature m the dense phase, heat delivered by the bed, and the thermal efficiency of the bed are larger when calculated by the Rx-Model This was to be expected smce the Rx-Model considers a highly exo- thermic homogeneous reaction m the ddute phase which in the simpler model (M-Model) had been neglected

The efficiency of the bed defined as

E_ C!m_ k-MT, - TV.1 Fo b-L&, (5 1)

mcreases also because the feed rate of particles F. 1s practically not affected by the choice of model As can be seen in the tables, the mflow of par&les necessary to

mamtam the steady state condltlon 1s strongly dependent upon the size of particles being fed as well as on the excess gas flow over that for munmum flurdlzatlon We had been m Ref [l] that d mass transfer at the particle surface 1s a hnutmg step, F. IS qven by

5WK F*=R’

I (52)

wlth

K = yf(2Cf1 + C;,) (53)

x, = --(I + UPM&X~ - (~2~1~ - 1)) + 5 w#(2,9meo + L, - Lax, + 5 W,DtpMTn” + L&, + HcT,.

(l+aP+H,) i4‘,)

718 A L GORLXDN et al

Tabk 1 Parameters values for calculation of steady states

dB Ri case m - oo41tf - - I 0 15 1 55 11 0 15 3 55 III 0 15 5 55 Iv 0 25 1 55 ” 0 25 3 55 VI 0 25 5 55 VII 0 15 3 55 VIII 0 15 3 55 IX 0 15 3 55 X 0 15 3 25 XI 0 15 3 75

To

l-K1

550 550 550 550 550 550 800 800 800 300 300

500 500 500 500 500 500 300 600

1,000 500 500

TI C’KI

Q-7 -6 x 10

wa+ts/m3

Q&*10-6

jO"ldm3 K9

348 1.101 0 670 6 931 827 1,106 0 529 2 303 880 1,116 0 499 2 041

674 775 827 827 827 598

1,010

994 934

1,155 1,171 1.178

-982 1,090

0 475 2 537 0 397 1 844 0 562 2 446 0 573 2 494 0 570 2 516 0 762 4 587 0 406 1 447

XII 0 15 3 12 5 300 500 1,496 1,161 0 298 0 693 XIII 0 15 3 25 800 500 598 1,050 0 841 5 063 XIV 0 15 3 75 800 500 1,010 1,226 0 478 1 703 Xv 0 15 3 12 5 800 500 1,496 1,354 0 354 0 852

500 1 I I I I I I, I,

2 4 6 6 IO 12 I4 I6 I8 20 L+,/U,,,f - Excess flow 0”~ Inclplent fhdization

FIN 1 Effect of U, and p on steady solutions (small bubbles)

where CT, and C2, are the concentrations of oxygen and carbon dloxlde respectively III the mterstrtlal gas because those values change when one or the other model IS applted, we would expect FO also to vary However, It was found that K does not change with a change of model so that F. remams unchanged for a gven set of input parameters Therefore, the thermal efficiency of the bed, as defined, depends exclusively upon the heat transfer between the bed and walls or submerged surface

(a) T Tf (61 T

1.600

I.500

I.400

I.300

E I,200

e = 5 I.100

2 1.000

I-” 2 900

fj 800

f

,"

700

600 I 3 5 7 9 II I3 I5 Uo/U,, -Excess flow over inciprmt

fluldlzation

Rg 2 Effect of Us and p on steady state soluuons

The multiphcity behavior can be compared by looking at the curve corresponding to T” = 300°K m Figs 1 and 2 and Ftg 4 of Ref. Ill From Fig 1 we observe that multtple steady states occur at a much larger ratlo of UdU,,,, and the range of thus ratio for which multiple steady states occur 1s larger than m the former case Because the system 1s a relatively complex one, It seems

FU 3 Heat generation and rejection curves

Alodelhng of fluldued bed reactors-v 719

there IS no sunple and duect explanation for the differences found usmg the two models However, we beheve that some mslght into multlphclty can be gamed d the bed ts viewed through a &fferent structure of the phases We recall that the dense phase was assumed to be perfectly mixed while we assumed that the dilute phase was m plug flow Besides, It was assumed that all excess flow over that needed for mnumum fluuldlzatlon passes through the bed m the form of bubbles, that IS, In the dilute phase Fmally, comparmg the values of T,, the temperature of the mterstmal gas with To, the tem- perature of the mflowmg gas, it IS observed that the latter 1s almost mvarrably much lower than the former All this suggests that the fluidlzed bed can be viewed, at least with respect to the heat transfer process, as a contmuous flow stirred tank reactor (represented by the dense phase) with a coolmg co11 Immersed m the reactor (the dilute phase) Although this IS In many aspects a very gross interpretation of the bed, it helps to develop an understanding of the multiphclty behavior m the same manner as in a typical continuous flow stured tank reactor (e g , AIIS[~]) Thus, we can compare the curves in Fig [4] of Ref [l] and Fig 1 of this paper by observing the diagrams m Fig 3 In the former we see that for values of Uo/LJ,,,~ smaller than approxunately 7 and greater than 9 5 there IS a umque solution, while m between there are three solutions This can he visualized from Fig 3(a) where the change m the ratio UdCJ,, translates itself in a change in the slope of curve B For higher values of UJU,,,, the hne becomes steeper and the intersection with the abscissa moves to the left producing a unique solution (wth a low temperature) For small values of this ratio the lme bends down and LS displaced to the r&t glvmg again a unique solution (with high temperature) Other curves presented can be explamed m a similar way For Instance, the curve T = 800°K m Fig 4 of Ref [l] Bves only one solution because a warmer coolant means a parallel displacement to the right of curve B In Fe 3 The Rx-Model IS more mvolved but still can be dlscussed m a slmllar way Including the reaction m the dilute phase means an addltlonal source of heat generation Furthermore,

Fu

I I I I I ,I I t I I t =I 3 5 7 9 II 131517I9212325

da Effectwe bubble dmmeterlcmsl

4 Effect of d, and R, on posslbie steady states

because Fe 1 represents a bed with small bubbles, the heat transfer coefficient between the bubbles and the dense phase is very he (see Fig 6 of Ref 113 and for expressions for the coefficient see Ref ISI) and hence we can assume that the heat generated m the dilute phase IS generated withm the dense phase Thus, as shown dlagramatically m Fig 3(b), the heat generatlon curve A’ becomes steeper and displaced upwards m comparison wrth curve A of Fig 3(a) It follows that for equivalent coohng condltlons (that LS, the same value of p as well as ratio lJ,,/U,,,~) gtven m curve B, a umque solution 1s obtamed Multiple solutions are agam obtained d the ratio UdU,,,, IS mcreased but that IS eqluvalent to m- creasing the cooling effect, mahng curve B steeper, as in I?’ Notlce that although the angle to be swept by the lme B’ m the repon of mulUphclty LS smaller than m the previous case, for steeper slopes we requue a much larger change m the slope for the same change of angle This would explain the widening of the multlphclty reDon

The sltuatlon 1s different m Fig 2 which used the same set of coohng parameters as m Fig 1 but for a different bubble size In this analysis of the bed the role of ds IS

,80_“lJNmt=s5 Rl * 0003n

co I

6 -I4 150-

2 ; e -I2 3 z 140- % r 0” 4 S

6 100

_r-; -IO ,‘ 120- r’

8 t -

___--------., 6 E z

) g OBO- - __--- 0

= d

/’ 1 060- I’

-4 =

3 z -2

2 c

-0 I:

I 3 5 7 9 II I3 I7 15 I9 212325 d.-Effectwa bubble dmmeier(cms1

FQ 5 Effect of d, on concentration and efficiency of combustor (homogeneous reactlon m bubbles)

”

040 c 1 ‘1

I 3 5 7 9 II 13t5t719212325 ds - Effective bubble dlatnefw (ems)

FIN 6 Effect of d, on concentratton and eteclency of combustor (no homogeneous reaction m bubbles)

the

the

720 A L &XLXlN

related to the beat transfer coefficJent between coolant aRd reactor Thus, a higher value of & dlrmnlshes the rate of heat transfer as well as the slope of heat genera- tlon curve Thus, both curves of FJ~ 3 are modified, and m a maRRer not obvJous for predictJon. In any event, the changes in behavJor between Figs 1 and 2 are due to the change m dB values, whJch will be reviewed m the next sectlon

INFLLIEBKX OF BUBBLE BlZE

As can be concluded from FJ~ 4, which is to be compared with Fig 5 of Ref [I], the bubble size plsys a much more important role m the multipkty behavior of the bed in the new model Thus was to be expected smce now the bubbles play an important double fun&on not only as a bypass for the cool au (If T” Js low) but as a source of heat because of the exothermic reactJon Jn that phase. For compmson we can take the curves for R, = 5 mm in the figures mentioned above It is clear that for bubble sizes above 5 cm there IS a sharp drop in the JntershtJal temperature, Although there is a reduction in the coolmg effect of the ddute phase, the mam effect 1s probably the decreased oxygen supply to the dense phase which Jmphes a reduction Jn the role of the carbon monoxide oxidahon

The values of the concentrations for a fixed value of L&/U,, are presented in Fig 5 It can be seen that the concentration of oxygen m the dense phase C,, remains almost constant for the range of dB studled But it LS clear that this IS at the expense of a reduchon in the rate of combustron of carbon monoxide as shown by the increase m the carbon monoxJde concentration and the decrease m the carbon dJoxJde concentratJon, gJvmg further support to the mterpretatlon gJven above for the reduction in the interstitial phase temperature In the same figure we can see that the efficiency of the bed as defined m eqn (51) shows a modest increase due to a reductJon Jn the amount of carbon being burned

EFPECT OF INTERCFIANGE COFSFFIW It has been shown that the Jnterchange coefiicwnts for

mass and heat transfer between the dilute and dense

phases have a strong effect not only upon the values of the dependent variables but in partJcular upon the mul- UplicJty behavior of the bed[3] In order to find this Jnfluence the coefficJents are calculated by means of two theories In the Kunu and Levenspiel model[6] we con- sider mass transfer resistance for both bubble-cloud transfer and cloud-emulsion transfer whale Jn Davidson’s model only the first resistance 1s considered For heat transfer only bubble-cloud resistance was consJdered relevant Jn both cases

The Jnfluence of the coefficJents can be seen in FJg 7 and comparison with Fig 1 which results from the same model and set of parameters In both cases the net effect of mcreasJng UdU,, IS the same the combJnation of Jncreased cooling effect and enhancement of the com- bustJon drives the temperature of the gas through a maxJmum For even larger values of L&/U,, the coolmg effect predominates so that the reactor cook off However, withJn the range of parameters WV,, con-

Uo/Umt-Excssr flow over mc,pmn, ttu,d,zot,an

FIN 7 Effect of Interchange coefiiclents and L&/V,,,, on posstble steady states

d,- Effectwe bubble dmmeterkmr)

FIB 8 Effect of rnterchange coefficients and ds on possible steady states

sldered here, no sudden translhon to a quenched state IS observed Furthermore, a very different behavJor be- tween relatively bJg partJcles (R, = 5 mm) and smaller ones (R, = 1 mm) is observed In the case of large park cles the temperature of the mterstltial phase will react a maximum of approximately 1900°K when l&/U,, = 5 and then decreases gradually wJth an Jncredse Jn the flow to the bed In the case of the smaller particles the temperature increases with the flow to 900°K when the iiow JS UdU,, = 5 and from then on IS hardly affected by the ratio U,JU,,

In Fig 8 we can observe the effect of dB on the solution behavior when Davidson’s theory is used to calculate the mass transfer coefficient It should be compared wJth Fig 4 (and Fig 8 of Ref [l]) whrch are the results for the same set of parameters but wJth KunJJ and LevenspJel values of the mass transfer coefficrent We had seen that for those cases the sJze of bubbles had a very sharp effect both on the multJplJcJty and upon the temperature of the gas in the dense phase On the other hand, from Fig 8 it can be seen that when Davrdson’s theory Js used the effect of de IS almost negligible and that the multJpltcrty observed Jn the M-model has dJs- appeared In fact, for this case the rate of interphase transfer IS so hJgh that the temperature and multip1icJty behavJor depend maJnly upon other parameters (for m- stance the sJze of partrcles. as can be seen Jn Fig 8) The above findJngs may be summarized by sayJng that mul-

Modellmg of Bmdued bed reactors-V 721

ttple steady states are less probable for higher rates of transfer between phases

NlJMERlCAL COMPUTATIONS OF G-MOD&L COMPAUISONS

In Ftg 9 we have plotted steady state results for the three models studted, that IS the M-Model (see Ref [l]), the Rx-Model analyzed tn a previous section and the G-Model correspondmg to the general case It IS obser- ved that the results of the Rx-Model are mtermehate between the two other models The same pattern of behavior can be observed m the followmg results It should be notrced that the values of the M and Rx- Models tend to converge as the ratlo UJU,,,, goes to small values, the reason being that at &IV,, = 1 there IS no flow over mclplent lhudlzatton and, hence, no bub- bles Thus, both models must comclde stnce their only difference conststs of an exothermlc reaction wlthm the bubbles It IS also clear that the M-Model underestlmates the thermal eflklency of the bed because It neglects the exothermlc reaction occumng withm the bubbles This effect IS much sharper m Fg 10 smce m that case a good part of the bed consists of bubbles As could be expec- ted, the Rx and the G-Models tend to lpve the same results as the temperature increases and the rate of the reactions approach the dlffuslon luntt postulated for the Rx-Model

In Fig 11 we observe that agam models M and Rx

i

dg-O05m

RI-0003m T-I 300-K

2w Tp’. I.ooO-K

z 1.900 -

-L e

1,700 -

g ii 1500 -

5 1.300 -

% 3 I.100 -

,o - J 900

b 700 -

500 I 1 * . 1 8 1 lo iJ 0 2 4 6 6 IO I2 Uo/Umf - Encess flow incipient fbuidization

Fu 9 Effect of superlicral vetoclty Companson of models (small bubbles)

Y-M0d.I

sool 8 ’ ’ n ’ ’ ’ 0 ’ 8 8 II 123456789 IO II 12 I3

Uo/Umf - Excess flow over I”CIPI~O~ flu~dorotton

Fig IO Effect of superthaI veloctty Comparison of models (large bubbles)

RI-0 003111 T* .300-K Tp,--c.ooo-K

“o/an‘ - s s

3 5 7 9 II 13 15 17 IS 21 23 25

d,- Effectwe bubble dwneterlcms)

Fig 11 Effect of bubble diameter Comparison of models

d,=O osn I@00 - RI -0 003m

z 1.600 - lz.n.ro, F&t5 .

2 1.400 - m‘fuelon cim - Ran .? ~I.200 -

I 1.000 -

/ ol~urion Urn - No Ran

g 800 -

-F 600 - %

,_ 400 -

200 400 500 600 700 a00 900 1000

Tp’ - Dlmenahol particle feed tompwalure PK 1

Fa 12 Effect of psrtlcle feed temperature Comparison of models

underesttmate the thermal efficiency of the bed and the difference IS about constant for mcreasmg values of dB The dtierence in values IS dlstorted at very low values of dB but these values of dB are very unreahstrc In par& c&r, the predicted values of interphase coefficients become extremely large and the vahdlty of the models doubtful

In FQ 12 we have plotted the comparative effect of the particle feed temperatures on the steady state values of the system for a mven set of parameters For the cases where dtffuslon of reactants IS a hmltmg step, with and without homogeneous reaction m the bubbles (that LS, Rx and M-Models respectively) there 1s practically no influence of T,” This is to be expected d we recall that the sunplticabons made on those models Imply that the fimte rate of the heterogeneous reacttons IS being neglected In other words, smce the rate of reaction IS a function of the particle temperature, It IS clear that the sunphfications do elnnmate the possible mliuence of this vatrable For the general case, that 1s G-Model, the results are sun&u In nature for some range of T,” Nevertheless, for low values of the vmable the bed remams in a quenched state, in fact, it does not Igrute, particularly If the temperature of the milowmg gas is also low (as was the case m Fig 11 smce 7”’ = 300°K) It should be pointed out that the curve for the G-Model decreases slowly for increasing values of T’,.,O whtie the one for the Rx-Modet increases slowly If the assump- tions made hold perfectly the two curves should coincide for large values of T,”

By examuung previous results as well as the ones lust

722 A L GORDON et al

obtamed, some trends can be observed In general, both sunphfymg models, M and Rx, underestimate the poten- tlal heat and efficiency of the bed The difference IS a moderate one with the Rx-Model and wouId be very high with the sunpler M-Model On the other hand, we can tentatively conclude that the results for the Rx and G-Models become close as the particle feed temperature and the ratio QJU,, increase

NoTATloN

Symbols appearmg m the text, but which are not listed below are defined m Ref [ 1 J

Jacobian used m Newton-Raphson lteratlon constants defined m Ref 131 effective bubble diameter thermal efficiency defined by eqn (51) Integrals defined by eqns (19)_(22) defined m eqns (28) and (31a) temperature of mterst~tnd gas temperature of particles m a feed superficial gas velocity vector of concentrations and temperature m

the mterstitml gas vector of concentratrons and temperature m

the dilute phase

2 ax4 distance measured from the bottom of the bed

Greek symbols a,, stolchlomemc coefficient of component J m

reactlon 1 e,, varuttional of y with respect to x defined by

eqn (31) v, varIational of T,(r) with respect to x defined

by eqn (32) 01, ~3 functions appearmg in eqns (36) and (37) and

defined exphcltly m Ref [3] & vanahonal of R(r) with respect to x defined

by eqn (33)

REwIlENCES [l] Gordon A L and Amundson N R , Chem Engng SCI 1976,

31 1163 [2] Davldson J F and Harmon D, Fkrdued Purttcles Cam-

bridge Umverslty Press, Cambridge 1963 131 Gordon A L , Ph D thesis, Umverslty of Mmnesota (1975) [4] Ans R , Introductmn to the Anaiysrs of Chemrcal Reactors, p

155 Prentice-Hall, Englewood Chffs, New Jersey I%5 [5] Kunu D and Levensplel 0 , Fkdlzatron Engmeenng Whey,

New York I%9 [6] Kuml D and Lcvensplel 0, Ind Engng Chem Proc Des

Develop 1948 7 481 [7] Brown K M , Proc Pittsburgh Conf Num Anal (1972)