Development of a Model for kinetics

8/7/2019 Development of a Model for kinetics

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Techniques for developing a kinetic model of the codimerization of propylene withbutene over a heterogeneous catalyst are considered. The importance of using selectivityrelationships to simplify the modeling task i s demonstrated. Statistical arguments areused to show that the rate constants for propylene and butene dimerization are relatedto their codimerization rate constant by their geometric mean. Finally, an eight-parameterkinetic model i s proposed as describing this system. Experimental data are used toconfirm the model.

above. The authors are not in terested in present ing thebest k ine t ic model but ra ther techniques for modelbui lding. The approach employed takes into account theovera l l funct ioning of the ca ta lyst - reac tant system whi lepreserving basic kinetic concepts. The codimerization reac-tion of propylene with the isomers of butene over a het-erogeneous ca ta lyst i s used to i l lust ra te these techniques.T he ul t im ate use of a model will influence th e simplifyingassumpt ions which can be made . In t h i s s t ud y , t he au thor swere interested in developing a m odel which would predic tproduct se lec t ivi ty and conversion as a funct ion of tem-pe ra tu re, p ropy lene - t a -bu tene feed r a t i o , and t he amoun t sof the various butene isomers present in the feed.In the first section, some simple concepts of selectivityare used to de termine accura te ly the re la t ive ra tes ofthe important reac t ions and to provide considerableinformat ion about the reac t ion system. Some e lementarysta t i s t ica l a rguments are then given to show how thedimeriza t ion and codimeriza t ion ra te c onsta nts a re re la ted.Ne xt , the formula tion of th e kine t ic m odel i s considered.Judicious grouping of compound types is used to reducethe nu mber of te rm s in the kine t ic m odel. The se lec t ivityre la tions are used to reduce the nu mber of con stants tobe de termined by a regression technique . Fina l ly , da t aare presented to i l lust ra te the pra c t ica l i ty of th is appro achfor the reac t ions be ing studied.Initial Selectivity Relationships

Before the detailed kinetics of the reaction were consid-ered , examination of the pro du ct selectivity relationshipsprovided qu i te a bi t of informat ion. I f produ ct concentra-tion is plotted as a function of conversion for severalexpe r imen ta l runs , t he da t a sca t t e r much l ess t han ifthe corresponding concentra t ion vs. t ime data are plot ted.Th is i s because in se lect ivity p lots the var ia t ion in ca ta lystactivity is, in effect, canceled out. Also, product selec-t ivit ies provide means of de terminin g the re la t ive ra tesof reaction. This information is quite useful in developinga m ore comprehensive kine t ic model.Plots of concentra t ion vs. conversion da ta f rom f ivese lec ted ba tch autoclave runs were made (only one ofwhich i s shown here) . E ach of these runs was made w i tha di f ferent in i tia l ra t io of b utene:propy lene . T hus , i t waspossible to s tudy how se lec t ivi ty var ied wi th feed ra t io .In the feed for each run, the ra t io of butene-2 to butene-1was held constant a t 2 .5 . This i s impo rtant s ince butene-1 i s substant ia l ly more reac t ive than butene-2 and hasa significant influence on the product selectivit ies. A t y p i -cal concentration vs. conversion plot is shown in Figure1.Th e slope of th e con centra t ion l ines a t zero conversionare th e init ial selectivities;

S , = limit ~ (1)i = 6, 7 , 8

whereX = conversion of feed olefin weight fraction, 1 - [ (CJS , = initial selectivity of hexenes ( i = 6) , hep t enes ( i= 7 ) , and octenes ( i = 8)C, = weight frac tions of propylene ( i = 3) , bu tene ( i= 4 ) , hexenes ( i = 6) , hep tenes ( i = 7 ), a n doctenes ( i = 8)The init ial selectivit ies have significance because theyi l lust ra te how th e f i rs t molecules appea r to combine . I fwe assume tha t a t very low conversions the reac t ionson the ca ta lyst surface are

+ C4) / (C 3 + C4) lnltlal I

c?,+ c4 5 c;k s2 c4 - C8

and tha t format ion of molecules wi th a carbon numberof 9 or higher is negligible we have the following,dCs = k6C; dt ( 2 )

dC i = kjC,C,dt (3 )dCs = k8C:dt (4 )

(5)

(6)( 7 )(8)

Conversion is r e l a t ed t o t he appea rance of produc t s byd X = dC6 + dCi + dC8

Sg = k s c i / (kEC3L + k7C3C4 + ksC4L)S i = k7C4CJ/(kfiCi+ kiC4C3 + ksCa)Sa = kgCi / (k6C32 + kiC4C3 + ksC4L)

and , t hus

Dividing the nu merators an d denom inators of the previousthree equat ions by C3C4:Ss = ( k s / R o ) / ( k s / R o+ k ; + k&o) (9)

201 I I I

I I I0 10 20 30 40 50 60CONVERSION, WT %Figure 1. Concentration vs. conversion, Ro = 1.5 (mole),C4 -2/C4- 1 = 2.6, 1 = 249"F

Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 2, 1971 2 51


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v,W2 0.8WFCLw > 0. 7=kg r 0 .6E d 0 .5E l 0.4IS85 0. 3

>0W

zcn

; 0.2ULT 0. I0.0

/// iI0.1 0.2 0.3 0.4 0.5 0.6 0.7RECIPROCAL MOLE FEED RATIO

( C4 I C 3 1-1Figure 2 . Ratio of hexenes to heptenes initial selectivityvs. inverse feed ratio

K ,K- = 1 5 8

S ; = k ; / ( k s / R o+ k ; + ksRojSs = k&o/ (he/Ro + k; + &Ro) (10,(11)

where R , , is the ini t ia l ra t io of butene -to-propy lene ina batc h reactor or the feed rat io to a plug flow reactor ,Ro = C,/C3 (12,

Note that these are nonl inear relat ions in R , , . However.by dividing SGan d Ss b y S - we getS61S:= ( k 6 / R o ji k ; (13)S e / S ; = ksR o /k ; (14 )

Thus, we can determine k , i k : a n d k q ; k : from l inear plotsof Sti/Srvs . R,-] an d SS!S: vs. R o , respect ively. Figures2 a n d 3 are good evidence that these basic linear relation-ships are veri f ied by the data (part icularly s ince al l dataare based on ini tia l s lopes) .The express ion for the ini t ia l heptenes select ivi ty canbe modi fi ed by d ividing the nu mera tor and the denomina-tor by k ; . T h u s

Si = l i {[ ( k e / R o ) / h ; ]+ 1 + (kaRo/k; j I (15 )Therefore, once k 6 $ k ; a n d k * , , k ; are determined fromFigures 2 and 3, we should be able t o calculate the initialheptenes selectivity as a function of feed ratio. Figure4 shows the calculated and actual ini t ia l heptenes selec-t ivi ty vs . feed rat io. No te th at t he measured select ivit iesare consis tent ly less than the calculated values . T h i s isbecause we assumed no formation a t low conversions ofmolecules of carbon number greater than eight (Cn+).In fact . C,+ formation does occur to some extent ( - 5t o 10'; selectivity) at very low conversions. and its inclu-s ion would tend to el iminate the small discrepancy betweenthe calculated and actual values of initial heptenes selec-t ivi ty.I t i s interes t ing to know the ini t ia l feed rat io whichwill maximize the initial heptenes selectivity. Differ-

ent iat ion of Equat ion 15 with respect to R , , shows tha tthe maximum heptenes selectivity will occur at a feedratio given by(16)R,,@ M ax S-= \ (k , !h-1 ( k ~ : h ; ) ] '

The maximum heptenes select ivi ty at this feed rat io isS: M a x = 1 [l + 2 ( k , h- i l ' (hG h - i ' . ] (17)

For the butenes feed used in this s tudy. a ma?rimuminitial heptenes selectivity of 19.1$ will occur at a feedratio of 3.05 mole. Thus . th e maximum h eptenes selectivi tywould appear to be l imi ted to about 5 07 .Statistical Relations in Codimerization

>

The most important react ions in this sys tem involvethe reaction of propylene with propylene to form hexenes.propylene wi th butenes to form heptenes . and buteneswith butenes to form octenes . Since the catalys t has avery definite activity for the reaction of propylene withitself and a different but very definite activity for thereaction of butenes with themselves. w e were hopeful ofdeducing a relat ion between these act ivi t ies and theac t iv i ty of propylene reac t ing wi th bu tenes to form hep -tenes. T he fo llowing i s an engineer ing dem ons t ra t ionof th e expec ted re la tion . T hi s demo ns t ra t ion wi ll bedeveloped by considering the probabilities of tw o moleculesbeing adsorbed and in a react ive s tate on a catalyt icsite.

Consider first th e probabilities of rea ctan t moleculesbeing positioned opposite each other across a catalytics i te . For t w o reactant molecules. .4 a n d B (propyleneand bu ten e). the four poss ibi li ties are presented sch ema-tically.A S BA S AB S AB S B

(S is t h e a c t i v e s i te )

This representat ion is meant to show that i f a moleculeof A is positioned on one side of the catalyt ic s i te . i tis possible that a molecule of either A or B will be posi-

Figure 3. Ratio of octenes to heptenes initial selectivityvs. feed ratio

K * / K - = 0.17

2 5 2 Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 2, 1971


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Th e expected relat ion between rate con stants was tes tedon ba tch autoc lave da ta t aken a t three d i f fe rent t em-perature levels. 2 (h shs ) i h - was 1.15, 1.04, a n d 1.02 a t204,245, and 285F, respectively. For perfect corrobora-t ion of theory with fact , the value should be uni ty foral l temperatures . This is excel lent agreement , especial lysince the hs are taken from s lopes.Th e relat ion between rat e con stant s should be usefulin reducing the n um ber of indepe nden t con stants requiredfor a reasonably comprehensive kinet ic model for thissys tem.The authors da ta showed a maxim um in i t ia l heptenesselectivi ty of about 50% Eq uat io n 15 will give a maximuminitial selectivity of 50% only i f Equat ion 33 holds . I tis s ignif icant that the formula relat ing h - to h s a n d hawas derived indepe nden t ly from the equa t ion for max imu mini t ia l heptenes select ivi ty. The fact that this formulais the only one which will maximize initial heptenes selec-t ivi ty at 50; is imp ortan t corrob orat ion.Kinetic Model

Consider now the development of a kinetic model ofpropylene-butene codimerizat ion.The first thing to do is to find a sequence of kinetics teps which accurately describe the data . This is a lmostalways an iterativ e procedu re of proposal of a model,f i t t ing the mode l to the da ta , and checking the fit a n dadjus t ing the model or taking more data as needed. Theses teps are repeated unt i l a reasonable kinet ic descript ionof the sys tem is obtained. T he model formu lat ion s tepis most im por tant , requiring experience and common sense.I t i s a t th i s s t ep tha t one can become t rapped by h i sown fundamental ism and propose more and more complexmodels which are cumbersome t o use and im pract ical toeva lua te . I t i s impera t ive a t th i s s t age tha t the mode lbe kept as s imple as poss ible while maintaining theimp ortant features to be described. Th is is why th e

modeler must have clearly in mind what the ul t imateuse of the model will be.After going through this i terat ive procedure one f indsthat the obvious react ions adequately describe the datafor th i s sys tem.

C, = concentration of propyleneC, = concentration of b u t e n e - 1C , . = concentration of butene-2C, = concentration of hexene

C- = concentration of hepteneC8 = concentration of octeneC9+ = concentrat ion of molecules having a ca rbon nu m-ber nine and greaterh , = ra te cons tant sHere the c i s and t rans butene-2 a re grouped toge theras one compound, C, - ? ; also all higher molecular weightt h a n C,s are grouped together as Cs+. This reflects thelack of primary interest in the higher molecular weightcompounds . React ions 34 to 39 more or less reflect theobvious kinet ics of the sys tem. The interact ion react ion,C, + C4 - + C8 was not included because i t was fel ti t would not material ly help the fit of the model to thedata . Equat ion 39 is jus t a very gross s implif icat ion ofthe formation of Cs+. E q u a t i o n 40 is the isomerizationequi librium reac t ion be tween b utene-1 and b utene-2 . N otethat the reverse rate constant is specif ied by the forwardra te cons tant and the known equi l ibr ium cons tant . Thelas t equa t ion was in t roduced to t ry to account for theobserved finite Cs+ selectivity at low conversion.(Eq uat io n 39 alone would predict a 0% selectivity toCq+ at 0% convers ion.) Again, this was an at tempt toaccount for something with our model in a very grossway because Cs+ modeling was not the prime object ivein the model.These eight reaction sequences generate the followingra te equa t ions :dC, /dt = -2 k iC; - k2C3Cq - i - k3C3C4 - 2 -

k&3(Ce + C.i+ C,) - 3 kgCg (42)dCq - l / d t = -k2C3C4 - 1 - 2 k,C: - 1 -

ksC4 - i(C6 + C; + C,) - kj.[Cd- I - (Cq - 2 / 1 2 ) ] (43)dC4 - z /dt = -h3CSC,- 2 - 2 ksC: - 2 -

ksC4-2(C6+ C 7 f C a ) + h . i [ C c - i - (C , -2 /12 ) ] (44)(45)dCs/dt = k iC; - ksCs(C3 + Cq - 1 + Cq - 2 )

d C i / d t = k2C3C4 - 1 + k3C3Cq - 2 -kGCi(C3 + C4 - i + Cq - 2 ) (46)dC, /dt = k,C: - 1 + ksC: - 1 -

k6C~(C3+ C4 - 1 + Cq - 2 ) ( 4 7 )dCs+/dt = ks(C.i + C, - 1 + Cq - 2 ) (C6 + C- + Ca) +

hgC3j (48)T h e 1 2 i n E q u a t i o n s 4 3 a n d 44 i s the equi l ibr ium cons tantfor but ene isomerizat ion. As can be seen, this is a setof seven coupled, nonlinear, differential equations for whicheight rate constants must be found to give the bes t fitto experimental concentration vs . t ime dat a . Conceptual ly,it is possible to directly apply some of the recentlydeveloped regression techniques (Seinfeld. 1970) to deter-mine th e eight rate constants in the model . T he avai labi l-i ty of a computer makes this qui te feas ible . However,i t has been th e auth ors experience th at direct appl icationof regression techniques to a set of data leads to seriousdifficulties. While a good fit of the model to the datawas obtained using the regression method of Jones etal. (1967), some of th e con stan ts determin ed w ere negativea n d / o r dtd not follow a consistent (Arrhenius) patternwi th t empera ture . Thi s may have been owing to inac-curacies in our data but we feel it is probably owingto the fact th at a n orthogonal set of d at a was not avai lable.Because of this it was found th at d irect , ra the r bl ind

2 5 4 Ind. Eng. Chem . Process Des. Develop., Vol. 10, No. 2, 1971


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0.06 I I I I

.oo 0.02 0.04 0.06 0.08C7, LB MOLE /F T 3

Figure 5 . Hexenes vs. heptenes, T = 240" Fapplications of regression techniques were not useful. Iti s a t t h i s po in t t ha t t he p re l im ina ry se l ec t i v i t y and s t a -t i s tica l concepts which ha ve been developed p rove to beinvaluable.Selectivity relations discussed earlier provide one wayto simpl i fy the regression task. I f Equat ion 45 i s dividedby Equation 46 for low conversions, we get

d C s / d C ; = kiC3LI (k2C3C4 - I + k3C3C4 - 2 ) (49)Also from Equat ions 45 and 47 ,

dC s / dC s = k i C i / ( k 4 C ; - I + ksC: - 2 ) (50)I f , in addi t ion, we use pure buten e-1 for feed, these reduceto

dC s / dC i = h i / & { C, /C, - I 1dC s / dC s = k i / k 4 { C B /C4 - I 1'

(51)( 5 2 )

This means if we plot C6 vs. Ci a n d C g vs. CS , for apure b utene-1 feed, th e s lope of th is plot a t zero conversionalong wi th th e know n feed ra t io should ena ble ca lcula t ionof k l / k , an d k l / k 4 . These plots a re shown in Figures 5and 6 . Wi th t he se r a t i os i n hand , Equa t ions 49 a n d 50,along with the init ial slopes of cg vs. c; a n d c6 vs . csplots for a feed containing a known mixture of C, ~a n d C, 2 , can be used to ca lcula te k l / k 3 a n d k , / k j . O nefur the r cons t r a in t i s added a t t h i s po in t wh ich i s t ha t ,

k2 = 2 (k ikd) ' ' ( 5 3 )k3 = 2 (k ik s ) ' (54)

and

This i s f rom the sta t i s t ica l na ture of th is reac t ion men-tioned earlier. Plots similar to Figures 5 and 6 we re madefor three di f ferent tempera ture levels and wi thin theaccuracy o f t he da t a t he r a t i os o f r a t e cons t an t s d idno t va ry wi th t emp e ra tu re . From these ca l cu l a t ions andplots the fol lowing ra t ios of ra te constants were found:k , / k l = 1.68, k , / k l = 0.2. k 4 / k l = 0.705, and k 5 / k l = 0.01.These man ipu la t i ons o f t he da t a r educed by four t henum ber of ra te cons tants which must be fit t o t h e d a t aby a regression. This reduct ion should not beunderestimated because the success or failure of a largeregression problem l ike this depen ds heavily o n such redu c-t ions. W e have , in effec t ,j u s t k l , k 6 , k ; , an d ks t o de t e rminein a regression.Seventeen exper im enta l runs were mad e in a ba tch reac-tor . In each exper iment six t o t en sample s, t aken a tdi f ferent t imes, were analyzed for the seven com pound s

0.06

IC)t 0.04\W-10zm-13 0.02

0.00 1 0.02 0.cC8,LB MOLE / F T

4

Figure 6. Hexenes vs. octenes, T = 240" Fof i n t e re s t . These runs we re made a t va rious t empe ra tu re s ,feed ra t ios, and reac tan t di luent levels .The four remaining ra te constants were fit t o t he sedata wi th the use of the regression method developedby Jones e t a l . (1967) . This method essent ia l ly reducesour task to a s imple l inear regression. For problems ofthis complexi ty this method is recommended.One Arrhenius plot resulting from this regression isshown in Figure 7 . T h e r a t e c o n s t a n t s a r e s u c h t h a tthe ra tes a re given in poun d mole per hour-fee t' of ca ta lyst .Mod el Prediction

The plots in Figure 8 show how wel l our proposedmodel f i t s the da ta f rom one of the ba tch exper imenta lruns. The acid test of a model for this process is howwell i t reproduces the selectivity vs. conversion re la t ion-ship. The se lec t ivi ty da ta for severa l ba tch runs and thepredicted model selectivit ies are shown in Figure 9. H e r ethe biggest deficiency of th e mod el is sho wn , i ts inabili tyto p red i c t C9+ selectivit ies a t low conversion. If a correc-tion is made for this, the Cs, C;, a n d CR selectivit iesg ive much be t t e r ag reement wi th t he da t a i n t h i s f i gu re .We fee l tha t to model proper ly the Cs+ a t low conversionwould require an unwarranted amount of e ffor t .Product se lec t ivi ty pa t te rns are mater ia l ly inf luencedby the ra t io of butene to propylene in the feed. Thesese lec t ivi ty pa t te rns are a lso dependent on the re la t iveam oun ts of th e butene i somers in the feed. Figure 10is a plot of c6/cS vs. feed ra t io . Both ba tch and pi lot -p l an t da t a a re shown. Superimposed on these da ta arethe kine t ic model predic t ions. This plot shows tha t toequalize Cs a n d C g in the product a feed ra t io of 2 to1 should be used for a buten e-1 feed an d a 3 - to-1 ra t iofor our eq ui l ibra ted feed. I t should be noted tha t forda t a i n t h i s f igu re t he m ixed-bu tene ba t ch run used b u t enetha t w as no t equ i l ib ra t ed . Th i s m eans t he se r e su l ts shou ldbe somewhere be tween the pure bu t ene -1 and equ i l i b rat ed -butene feed resul ts as i s shown.Feed ratio has a significant effect on conversion. Thefollowing pi lot -plant d a ta show this .

Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 2, 1971 255


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20Jw:'Icnt 0 \ I l o t000.1- 0.0025 0.0026 0.0027I /T ("K)- l I I I II I I IIO 20 30 40Figure 7. Arrhenius plo t of k l

K, = 1.68 K IK i = 0.2 K IK? = 0.705K IKj 0.01 K I

CONVERSION, WT %Figure 9.Product selectivities (selectivity vs. conversion)

I I

mm L\W-I0IZ 0.05

m m-1 -1

LL\Wd0.00-0 50 100 150TIME, MIN

I 1 1 I0.04 1 -I0.00o,02DW-1g 0.01 m-1m-1 0.000.00' L 5 0 'TIME, MIN TIME, MIN- 0.11 I I 1 I I 10 I 2 3 4 5 6C4:C3 IN FEED (MOLE RATIO)mI- 0.010

wLL\

0.010Figure 10. Product selectivity vs. feed ratio as a functionof butene feed and conversion

Mixed butene feed (not equ i l ib ra ted)II

A Batch, 30% conversion0 Batch, 35 % conversion0 Batch, 40% conversionV Pilot plant , 30 % conversion/X Pilot plant , 30 % conversion

0 Pilot p lont , 18 % conversionButene-1 feed

2P 0.005S Y 1 3

0.00)) 1 I I0 50 100 150TIME, MIN

Figure 8. Fit of model to da ta (concentration vs. time)propy1ene:butene-1 feed and a 34 0-1 propyl-ene: equi l ibrate d-buten e feed are about 4 o r 5 t o 1. Ourkinetic model also predicts this difference in reactivity.Conclusions

This paper dem onstrates some techniques which s implifykinetic model building. Selectivity plots and relationshipsare of great value in reducing the regress ion task. Theserelationships also provide quite a bit of information with-out requiring solution of the set of nonlinear differentialequat ions represent ing th e kinet ic model .

C d C ! Conv er si on , O h M ode l p r ed i c t i on, Yo3 : l4.25:l 3327 32.628.6

A 6% convers ion point drop was observed when thefeed ratio was increased while all other op erati ng conditionswere held constan t . Thi s agrees wel l with w hat th e kinet icmodel would predict.A mixe d-buten e feed was much less react ive th an apure butene-1 feed. F rom both p i lo t -p lant and ba tch runsit is known that the relative reactivities of 2-to-1

256 Ind. E n g . C h e m . Process Des. Develop. , Vol. 10, No. 2, 1971


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The authors be l ieve they have developed a prac t icablekinetic model of the prop ylene -buten e codimerization reac-t ion. I t i s encouraging th a t th e model ext rapola tes sowell to the equilibrated-butene feed cases (only two ofthe runs used in the model development had bu tene -2 present in th e feed) . This model i s most useful in p aperstudies and for screening ideas to be t r ied in the pi lotp l an t .Acknowledgment

The au thor s t hank P. L. T. Brian of M IT for hisst imula t ing discussions about t he s ta t i s t ica l na ture of th isreac t ion. They a lso thank D . T. Robe r t s fo r mak ing theexper imenta l runs needed for th is model developmen t andfinally, Esso Research Labora tor ies , Humble Oi l andRefining Co. , Baton Rouge, La . for permission to publ ishthis work .Nomenclature

C, = concentra t ion of butene-1 , lb mole / ft3C, = concentra t ion of butene-2, lb mole / f t3C6 = concentra t ion ofhexene , lb mol e / f t3

C3 = concentra t ion of propylene , lb m ole / f t3

C; = concentra t ion of heptane , lb mo le / f t C8 = concentra t ion of oc tene , lb m ole i f t C9 + = concentration of olefins of C9 and higher , lb mole /ft Jk , =K , =P =Ro =r , =s, =t =v =

reac t ion ra te constantsadsorption coefficientsprobabili tyfeed ratio, C 4 / C I ,l b bu t ene i lb p ropy lenereact ivi tyselectivityt ime , m invelocity constant

literature CitedBox, G. E. P.,Hill , W . J. , Technometrics, 9, 57 (1967).Jones , C. R . , H i m m e l bl a u, D. M ., Bischoff, K . B., I n d .Seinfeld, J. H ., Znd. Eng. Chem., 62 , 32 (1970).Eng. Chem. Fundam., 6 , 539 (1967).

R E C E I V E Dfor review April 9, 1970ACCEPTEDNovember 5 , 1970Presented at the A IChE Meeting, Chicago, I ll inois , December 1970.

Pilot-Plant Development of theSulfate Recycle Nitric Phosphate Process

Robert S. Meline, Henry 1. Faucett, Charles H. Davis, and Arthur R . Shirley, Jr .Tennessee Valley Authority, Muscle Sh oals, Ala . 35660

TVA has developed a modified process for producing ammonium phosphate fertilizer(28-14-0) on a pilot-plant scale. Calcium i s removed by precipitating it with ammoniumsulfate. Sulfate requirements are minimized because the precipitated gypsum i s convertedto ammonium sulfate which i s recycled. Raw materials used are phosphate rock, nitricacid, ammonia, carbon dioxide, and a little sulfuric acid, ammonium sulfate, or gypsumfor makeup. Most of the development work has been directed toward precipitationand filtration of gypsum from the extraction slurry, and conversion of gypsum toammonium sulfate liquor and the subsequent separation of this li quor from the by-productcalcium carbonate by filtration. The WA studies have identified suitable equipmentdesign and operating limits for the primary variables.

Processes tha t u t i l ize ni t r ic ac id ra ther than sul fur icor phosphoric for acidulation of phosphate rock usual lyare attractive economically, particularly in locations wheresulfur must be imported. Some of these processes requiresupp leme ntal sulfuric or phosphoric acid an d therefor eare only partially effective in decreasing dependence onsul fur as a raw mater ia l (Young, 1966) . Th e Odda -typeTo whom correspondence should be addressed

process that is widely used in Europe removes calciumni t ra te physica l ly f rom the ni t r ic ac id ext rac t by crysta l -l iza t ion a nd cent r i fugat ion or f i l t ra t ion to avoid need forsupplementa l ac id (H ignet t . 1966) . By-produc t amm oniumsulfa te solution f rom capr olac tam prod uct ion has beenused commercially on a on ce-thro ugh basis to precip itateexcess calcium as calcium sulfate, which is removed byfil tration (Piep ers, 1966). A full sulfate recycle processhas been proposed for a long time a nd a patent for suchInd. Eng. Chem. Process Des. Develop., Vo l . 10, No. 2, 1971 2 5 7

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Development of a Model for kinetics