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Effect of surface diffusion on the selectivity of catalytic reactions
DIFFUSION ON THE EFFECT OF SURFACE SELECTIVITY OF CATALYTIC REACTIONS HAROLD H KUNG and MAYFAIR C KUNG Cbemlcal Engmccnng Department, Northwestem Umverslty, Evanston, IL 60201. U S A (Recerved 26 ApnI 1977, accepted 8 October 1977) IS made OQthe sffect of surface ddhslon on the selcctnnty of a catalyst for a consccutivc reaction A + B + C m a WAR-nuxcd s&red tank reactor The catalyst IScomposed of an LIrcgon despersed on the fi rcgoo A~B~s~sumodtooccurontheeart~aandE~Contbe~replo~.~~nofEfromatoBpFocccrdrbotb by surface d14Fusmnand gas phase transport Inhence of the flow rate through the reactor. the crystslhte SIZ of a, and the loadmg of the catalyst on the sckctw~ty for C m the presence of surface dtiuslon arc dwcusscd Under otherwme Antical cowhons, salectwlty ISmcrcased by surface d&won The ophmum cond&on for the pmducbon of C ISalso discussed
Ab&m&-Analysis
1lurRonucTloN
The effect of dtiuslon on the rate and selectwlty of both homogeneous and heterogenous catalyac reactions has In heterogeneous catalytic long been recorpuzed[l41 reactions, Muslon of reactants and mtermedlates into and out of the catalyst pores has been shown to affect the observed rate, apparent activation energy and order If the effectiveness factor devotes from umty[l, 3,53 The effect of pore dtiuslon on the selectwlty has also been analysed The results m&cat& that for btfuncuonal catalysts on wh~h the dfierent components cataiyse Merent steps m a consecutwe reaction, there are optldlstnbution and composltlon of the ~~~nents[2,3,6] In spite of the many analyses made on the effect of gas phase ~HUSMMI, very few were made on the effect due to surface tiusmn m heterogenous catalyhc systems The analyst of Sohn u al showed that sun&u to pore difkslon, an effectiveness factor can be defined for surface dtiuslon[7] Measurements on dlftuslonal mass transport m porous structures mdkated that under sutable condtions, surface dtiuslon can become the predbmmant mechamsm (over 50%) of transport[8,9] Therefore, it is log& to assume that surface tiuslon can also afkct the observed sekctlvlty It ls the purpose of this paper to analyse the effect of surface Musion on the selectmty m a consecutive reaction A+B + C The first step of the reaction IS catalysed by one regron P of the surface, whde the second by another regmn @, such as what one would encounter on a brfundonal catalyst Although we model the surface as cvcles of Q dlstriiuted uniformly on a surface of @, the conclusions can easily be extended to one dxmensronal model surfaces, such as surface of Pt with umform step density The selectway wdl be drscussed wab respect to three parameters, the flow rate of reactants over the catalyst, the crystal&e slzc of the 0 rcplon,andtheloadmg(theratioofsreeof a tothatof 8) of the catalyst. It ISassumed that the reactloll #sstructureinsensitive, this u), the rateu per af&ve We are mdependent of the particle size.
In Section 2, the model and the mathematrcal formulation is presented Sunphfition of the model to the form used m the calculauon of selectivity is summarized m Section 3, and the results of the sum&&on follow m Section 4 The model assumes that the catalyst surface 1s made up of c&es of o phase of radms r* and sate densHy x., umformly &stnbuted on the fl phase of site density xs as shown in FGg 1 The centers of the a clrcks arc at a &stance of 2h apart For sunphc~ty, the areas not meluded m the cvcles are assumed mert. As shown in Appendu A, a more exact treatment that mcludes reactrmty of these areas results m mod&a&on of the equations but not the general conclusion Various processes on the a and fi phases are sumrn_mFe 2(a) A+B~scatalysedona,andB+C on @ B, once formed on a, can nugrate to fl either by surface dtiuslon or by desorption from a and readsorption onto # Surface mob&ties of A and C across the boundary are neglected but as long as A and C do not react on p and a, respectwely, theu mob&&s should not &ect our conclusion Rapid desorpt&on of C ISassumed The reactor is assumed to be ideal contmuous stxrred tank type such that the partml pressures of the reactants over the surface are uniform, and are equal to those m the outlet stream, as shown m Fii 2(b). Thrs assumptmn
HAROLD H
IO04
KUNC
unphes mfinlte gas phase tiuslvltles Only A IS present m the u&t stream, and steady state isothermal operation IS assumed Materml balance for A and B on (1 gives
and MAYFAIRC KUNG
e,u+ e*B =
1
(8)
and, m dunenslonless varmbles, (9) where
(2) and e,*+ e,"+e,u = i
(3)
P = g
Combmmg these equations gives, m dImensionless varrables, (4) where
(k$Bo
+ k!& + b=)
Q =h2kBB Da I@ 0
(10)
(11)
Solutions to the Bessel equattons (4) and (9) are, respectively, V = AJd
U) + B,Ko( U)
(12)
and U=V(M)f
Y = A&,(X) + B&o(X) (5) (6)
Simtlarly, matenal balance for B on 0 gves
atr=O
emB ISfimte
atr=h
!!!?A, dr
(W
(15) A
2
wtth the appropriate boundary condrtions of
de.=
(a)
Fig
03)
6
(a) Rate constPnts Vambles
B
C
for tbe various reaction steps (b) for the reactor
IOand K. are mod&d Bessel functions Equation (14b) says that the flux of 8 out of o by ddk%on must equal that tnto /3 Equation (15) results from the assumptton of local equtltbnum m the dlstrtbutton of B between P and j3, wtuch, however, does not unply equthbrmm wtth gas phase A dertvatton of tlus equatton ISgtven m Appenti B The eqns (12) and (13) with the constants detemuned by the boundary condlbons describe m full the condotions of the surface These conditions can be further related to the observed rates of production of B and C and the conversion through matenal balance equations on the reactor.
(16)
1005
EtTectof surface dflusww on the selectivity of catalyuc reactions
&** IS found, all other necessary parameters can be determmed For most prac&zai cata&ts, D is about lo-’ to lo-” cm2/sec[8, 91, h about 10V3to lo-’ cm, and the rate constants about ld to 10V2set-’ [lOj Thus the range of P - Q and M-N isfrom -10’ to 10”
Once
(17) and Flpo = F(Ai - A,,) - FC,
(18)
In pnnclple, the eqtions developed wdl descnbe the behavior of the system completely However, values must be asslgaed to a total of e-teen vanables to solve for Ao, B. and Co Fortunately, unth a few sunphfymg assumptions, the effect of flow rate, parhcle size and loadmg on the selectrvlty for C can be evaluated, w&out loss of generahty, v&h a total of e&t assumed dunenslonless vanables 3-c-m
assume that (I) bBc ) kB1#, and (u) La * kE& and khAn then the selectivity S for C, as grven by eqns (16) and (17), LS lf
4.llEsuLT~HMULATION Effectof j3owfafe
we
’
-
co -
F(Ai - Ao) = 2aTxJk
By definmg a dunenslonless flow rate .V( = 1;12aTx,&‘k&) and dlvldmg It mto the left hand side of eqn (21), we obtam
$2q_$)[g(!y)
A,-Ao
Q
- A@x*zl(X*)
x-
(g-+$jg$)
The constants &, A, and BB are related to the other parameters in eqn (19) through the boundary condltlons To calculate S, we need now only to define eight Qmensiotiess vanables, M, N, P, Q, Q, DadDaB, DBkfJD..k;“e and r*/h In the actual sunulatlon, the vanabIes P - Q (or M - N), (M - iV)/(P - Q), Q/(P Q). NW-N), n, D.,,G/D~,s, D,kf!./D,k& and r*/h are defined In this regrouped set of vanables, only N/(M - N) IS related to Ao, only P - Q and r*/lr to h, only r*/h to r*, only Q/(P - Q) and N/(M - N) to & The calculation was performed by fust combmmg eqns (14a) and (14b) to @ve
+
B@x*K,(X*)]
[(I-;)+_u*Z,(U*)] To approxunate the actual operatmg Gon&tions more closely, the curves are calculated for an arbitrary constant concentration of A III the mlet stream that corresponds to a value of lo-* for the term k/x=) (h2/D,) k&A, BY settw cK&‘,Jx,,k;b) = 10m3 arbltranly, we have
J%-&- a-40)= lo-“-
S2?rTx,,D,
10-3
>
(22)
t)lvldmg eqn (22) into eqn (21), we obtam, N-z:!
xeDa VtP) IdU*) (e&a* - QIP) - XJL VW) ZIV.W
(CB* - NIM) II
The dependence of selectivity on flow rate for a gven set of parameters 1s shown m Fig 3 ms figure IS obtamed by sunphfymg and rearrangmg eqn (16)
Q)&-[(l-$)+.JJ*Z,(U*))
9= 1U-*- 1O-3
- zl(t/(P))K,(X*)IKI(t/(P))
Zo(X*)+
Z~(t/(P))KdX*)IK,(~/(P))
Thus, with eqn (15), are then used to solve for flBB* to give B* = 8 B* 08 =
forfk= 1
= -b*V(b2--4ac) 2a
where a=H(l-a, b=Hn+(l-n)($+H$)-1 .=l-b($-H$)
-fora> +forficl
1
An umque flow rate can then be calculated for a p;lven set of dunensionless vanables Using eqns (17) and (19), (ZG,/2rTx,JL) and S can also be calculated Asslgrung dtierent values to the terms (&x=) (h’/D,) k& Ar and (xek?,Jxak8) only shifts the curves along the x axis Hnthout changmg the trend shown Ftgure 3 thus represents the observed selectlvlty as a function of flow rate for 3 different catalysts operatmg with the same inlet concentration of A As expected, at very low flow rate, or long residence tune, selectivity is h@ because the intermediate B produced has a high probability of readsorption onto j3 for further reaction At high flow rate, or short residence tune, B produced IS quickly removed from the reactor, resukmg in a low concentration of B and low selectlvlty The ddferent
I
r
.
-4
.
-2 log
3
0
I
dtifuston and thus the effect of crystalhte stze should be most nouceable F-e 5 shows that seiecavtty IS constant for small crystalhte stze (small P-Q), wbd rt drops off at large crystalhte srze Tins behavror IS due to the fact that se~ecttvny depends on the ratto of the coveraKe of B on o and fi At low P-Q, (tLn/fbB) at the mterface IS almost constant whtch results m a constant S With proporttonaUy faster mcreasmg P - Q. 8,” maeas+ than @,“, and causes S to decrease
Effect ofload& Although loadmg
2
pt 3 K&et oi now rsts on selecuv1ty curves sre calculated with tg~fi) - 10, c~tm.$) = 10-3. a = ld, (fvlr) = OlmdM-N=IO-4 values of (M - NIP - Q) correspond to Merent rattos of (&J~=) The larger ts tlus ratlo, the lower 18 the dame rate of reaction of B, and correspondmgly the lower is S Relative rates of productron of c as calculated from eqn (17) for the condutons rn Fi 3 Is proportronal to the flow rate at low dew rate, and reaches a maztmum at high flow rate ‘Plus shtft from Lear dependence to non-dependence on flow rate occurs m the regton where the selecttvrty c-s from umty to 8 constant value Frgure 4 shows the frachon of C produced from sM= ,““” equals 2uTrD.. (d6-‘?dr)[,_,* The . . Wibutmn mxeascs wdb mcreastng flow rate, and becomes the only mechamsm of transport of I3 at veryhtphfiowrate Efixt of crpJtallrtc sue The effect of crystalhte stze of a(+) was studted assummg that the reactton ISstructure msenstttve, that IS, the rate constant per sue IS mdependent of crystallite stze The calculation IS done at a constant loadmg (p/h) and constant rat108 of (N/(M- IV)), (a(P - 0)) and ((M - N)/(P - Q)) The selectrvlty ts then evaluated as a functton of P-Q (1 e h) We choose to study the condWon at very hgb flow rate when dtifuston Is the only transport mechamsm for B to get from u to /3 Under tlus condthon, selecttvtty 1s zero m the absence of
IS generally defined as the we& percent of the a @base, It IStaken here as the fractton of surface covered by Q, that IS, (r+lh)’ The calculated values of S and the relattve yrefds of C are plotted as funcuons of (f/e) m Ftgs 6 and 7, respecttvely They are calculated for the condittons of very high flow rate such that dtffuston IS the only transport mechamsm, of small P - Q such @at there IShttle crystalhte stze effect, and of constant h We have also used the same set of parameters as m Frg 6 for the points at (9/A) = 0 1, and calculated a plot of S vs +/!I by varymg R and keepmg r* constant The resultug plot IS superunposable on Ftg 6 Physically, a constant h means a constant total surface
rn
0
-4
-2
2
0 log (P-d
Cwv~s are CdFu 5 Effect of crystalhte SIZCon se!echwty culated wrth &DJ.@p) = 10, (D,&LlDAT~ = lo-‘. fl= lw, (r*/h)=Ol,(N/(M-N))=OlandQaO
I
I
I
IO
l-o-
v)
0
Oo -4
4 Contrhubon
-2_
0
log2
2
-
from dhs~on for the production of C vahles of p8rameters are the same as 111Fig 3
01
02
03
w
05
06
07
06
P Fw 6 Effect of loadm on selechvlty Curves are calculated with I&D&@,,) = 10. (D&-/D&J = lo-“, f-3= Id. P - Q = lo-‘, (IV&W-N))=01 and Q-O
Effect of surface dtffuslon on the seltcttvtty of catalytic reactions
Fii
7 Relattve yteld of C as a function of load1118 Values of parametersare the same as m FJii 6
area of catalyst, and a constant r* means a constant area of a S mcreases with decreasmg loadmg m Ftg 6 because of two effects Ftrst. as (P/h) decreases, the ratto of the surface area of P to the length of the boundary decreases Smce desorptton rs proporttonal to the surface area, whde dtffuston to the length of the boundary, the relattve mportance of dlffuston mcreases as (r*/h) decreases, and subsequently S Increases The second effect is that the relative area of @ Increase wtth decreasmg (r*/h), whtch ISeqtnvalent to mcreasmg kgBc Thrs results m a decrease m the average coverage on B, whtch enhances the drtvmg force for dtiIuston, and results m an tncrease m S Although S mcreases with decreasmg loadmg, the rate of productton of C vartes ddferently wtth loadmg Figure 7 shows the optunum loadmgs at various condutons for the rate of productron of C To the rtght of the optimum, the reactlon on B IS hnutmg and the yield decreases wrth decreasing area of B To the left of the optuuum, the reaction on a 1s lmutmg and the yteld decreases wtth decreasmg area of a
1807
catalysts with and wtthout surface dtffuston IS less stratght forward In general. tt can be concluded that under otherwise tdenttcal con&tons, surface dtiuston enhances the yteld We have assumed m the analysts that gas phase dtiuston IS mfuutely fast A more exact model would consider the htgher concentratton of the mtermedtate B over a than over @ Such concentratton gradient 1s due to a fimte reststance to gas phase transport, and its presence would only enhance the effects due to surface dtffuston On a practical catalyst, crystal&s of a of some distnbutton of sizes are randomly dtstrtbuted on the j3 surface 1113 We expect that thts would only modtfy the detads of our analysts wrthout afIectmg the general conclusions Also, modtkatton of our two henstonal model to a one dunenstonal one, such as that for stepped single crystal surfaces can easdy be made We also expect that the conclusions here can be quahtattvely extended to these smgle crystal surfaces NOt’ATlON
mlet concentratton of A Ao, Bo, Co outlet concentrations of A, B and C D surface dtiustvtty F flow rate through reactor h radius of /I regton radtus of a regton k’ rate constant per site T total number of a regtons m the reactor n equthbrmm dtstrtbutton of B between a and B region 8 fractton of surface X sue density A,
Subscnp ts
-1 desorptton process 1 adsorptton process a process on a regton B process on /3 regton
5 coNcuIsIoN
The analysts presented mdtcates that surface dtffuston can greatly afkct the observed selecttvity of a reaction The presence of surface diffuston reduces the dependence of selecttvtty on the flow rate In fact, tf surface drffuston of B 1s much faster than its desorptton, a selecttvtty of unity. mdependent of flow rate, can be obtamed Comhtrons used m ultra htgh vacuum studtes (hke molecular beam expenments) are equivalent to the extreme of very htgh llow rate This analysts shows that dependmg on the degree of surface dtffuston, results from such studies may or may not dtffer srgtuficantly from the conventional &u&es Another mteresttng result due to surface dtiuslon IS the dependence of selectwlty on the crystalhte sue No such dependence would be expected w&out surface Mumon Surface dlfEuslon also results IIIthe increase m selectlvity vvltb decreasing loadThe selectivity approaches tuuty at very low loading. as shown 111FU 6, but would approach zero 111 the absence of surface d&tston Comparison of the effect of loadmg on the yield of C on
Superscnpt * value evaluated at the boundary between a A B AB BC V
andB process mvolvmg species A process mvolvmg species B reaction A + B reactton B + C vacant site
[ll Wetsz P B and Prater C D , Adu Cut 19546 143 (2) WheelerA , Adv Cat 1951 3 258 133Thomas J M and Thomas S J , Intmductlon to Zfeterogeneous Catalysts Academtc Press, New York 1967 [4] Rys P , Act Chem Res 1976 9 345 [S] Satterlield C N and Sherwood T K , Role of Lb#wo~ WI Catalysrs Adduon-Wesley, Readmg, Massachusetts 1965 161 Shadman-Yazd~ F and Petersen E , Chcm Engng Sa 1972 27227 [73 ~JI , Memll R and Petersen E , Chem Emgng Scr 1970 [8] SchnelderP andSmlthJ.AIChEI 196814886 191 Reed E M and Butt J B , J Phys Chem 1971 75 133
HAROMI H KUNG and MAYFAIRC KUNG
1008
tutls for the species must obey the equation
PO1 Baetxold R and Somorjal G , J Cat 1976 45 94 Ill] Luss D, I Cat 1971 23 119 APPEMNXA Actw~fy on the area of surface outstde the cwcles The centers of the cucles form a net of lattice pomts of hexagonal symmetry There IS one cucle m each of the unit cells of area 4h* sm (a/3) Thus the area per umt cell not covered by that the boundar; the circle IS 4h* sin (?r/3)- ?rh2 Akmmg condltlon of (d&‘?dr) = 0 at r = h IS stdl true, eqn (17) can be modtfied to
o=PS.+&l&,-~~--BsI
We now substitute the expression c = to + RT In (8/P) mto (A2), and choose as the standard state the condltlon when the adsorbed E IS m eqmhbnum with the gas phase B, we obtam
and
- eA).e,B = (I- en - eA]mi9,B’ eg9e.B ( I- eeB9emB’
(I - P ( 1-
- ?rh‘) keBc&’ I*
(AlI
(~42)
(A3)
But for the standard state, we also have
The equation for selectlvlty (eqn 19) must also be correspondingly modified
(I-
eB - tP ), ktB,, = kS,,BB’
and Equdlbnum drstnbutlon of B at the boundary Treat the dfluslon of B as a chemical reactlon sp+(B
S).=S,+(B
S)e
where S,, S, represent vacant sites on a and /3, and (E S) represents an occupied site At eqmhbnum, the chemical poten-
(1 -e,?‘) k:,Bo= k!,#’ Substltutmg these, and an expression for emAfrom eqns (1) and (3) into (A3). we get eqn (15) The assumption of local equibrmm IS equivalent to the assumption that the chemical potentml of the moMe species (B and vacant sttes) are contmuous at the boundary
Ab&m&-Analysis
1lurRonucTloN
The effect of dtiuslon on the rate and selectwlty of both homogeneous and heterogenous catalyac reactions has In heterogeneous catalytic long been recorpuzed[l41 reactions, Muslon of reactants and mtermedlates into and out of the catalyst pores has been shown to affect the observed rate, apparent activation energy and order If the effectiveness factor devotes from umty[l, 3,53 The effect of pore dtiuslon on the selectwlty has also been analysed The results m&cat& that for btfuncuonal catalysts on wh~h the dfierent components cataiyse Merent steps m a consecutwe reaction, there are optldlstnbution and composltlon of the ~~~nents[2,3,6] In spite of the many analyses made on the effect of gas phase ~HUSMMI, very few were made on the effect due to surface tiusmn m heterogenous catalyhc systems The analyst of Sohn u al showed that sun&u to pore difkslon, an effectiveness factor can be defined for surface dtiuslon[7] Measurements on dlftuslonal mass transport m porous structures mdkated that under sutable condtions, surface dtiuslon can become the predbmmant mechamsm (over 50%) of transport[8,9] Therefore, it is log& to assume that surface tiuslon can also afkct the observed sekctlvlty It ls the purpose of this paper to analyse the effect of surface Musion on the selectmty m a consecutive reaction A+B + C The first step of the reaction IS catalysed by one regron P of the surface, whde the second by another regmn @, such as what one would encounter on a brfundonal catalyst Although we model the surface as cvcles of Q dlstriiuted uniformly on a surface of @, the conclusions can easily be extended to one dxmensronal model surfaces, such as surface of Pt with umform step density The selectway wdl be drscussed wab respect to three parameters, the flow rate of reactants over the catalyst, the crystal&e slzc of the 0 rcplon,andtheloadmg(theratioofsreeof a tothatof 8) of the catalyst. It ISassumed that the reactloll #sstructureinsensitive, this u), the rateu per af&ve We are mdependent of the particle size.
In Section 2, the model and the mathematrcal formulation is presented Sunphfition of the model to the form used m the calculauon of selectivity is summarized m Section 3, and the results of the sum&&on follow m Section 4 The model assumes that the catalyst surface 1s made up of c&es of o phase of radms r* and sate densHy x., umformly &stnbuted on the fl phase of site density xs as shown in FGg 1 The centers of the a clrcks arc at a &stance of 2h apart For sunphc~ty, the areas not meluded m the cvcles are assumed mert. As shown in Appendu A, a more exact treatment that mcludes reactrmty of these areas results m mod&a&on of the equations but not the general conclusion Various processes on the a and fi phases are sumrn_mFe 2(a) A+B~scatalysedona,andB+C on @ B, once formed on a, can nugrate to fl either by surface dtiuslon or by desorption from a and readsorption onto # Surface mob&ties of A and C across the boundary are neglected but as long as A and C do not react on p and a, respectwely, theu mob&&s should not &ect our conclusion Rapid desorpt&on of C ISassumed The reactor is assumed to be ideal contmuous stxrred tank type such that the partml pressures of the reactants over the surface are uniform, and are equal to those m the outlet stream, as shown m Fii 2(b). Thrs assumptmn
HAROLD H
IO04
KUNC
unphes mfinlte gas phase tiuslvltles Only A IS present m the u&t stream, and steady state isothermal operation IS assumed Materml balance for A and B on (1 gives
and MAYFAIRC KUNG
e,u+ e*B =
1
(8)
and, m dunenslonless varmbles, (9) where
(2) and e,*+ e,"+e,u = i
(3)
P = g
Combmmg these equations gives, m dImensionless varrables, (4) where
(k$Bo
+ k!& + b=)
Q =h2kBB Da I@ 0
(10)
(11)
Solutions to the Bessel equattons (4) and (9) are, respectively, V = AJd
U) + B,Ko( U)
(12)
and U=V(M)f
Y = A&,(X) + B&o(X) (5) (6)
Simtlarly, matenal balance for B on 0 gves
atr=O
emB ISfimte
atr=h
!!!?A, dr
(W
(15) A
2
wtth the appropriate boundary condrtions of
de.=
(a)
Fig
03)
6
(a) Rate constPnts Vambles
B
C
for tbe various reaction steps (b) for the reactor
IOand K. are mod&d Bessel functions Equation (14b) says that the flux of 8 out of o by ddk%on must equal that tnto /3 Equation (15) results from the assumptton of local equtltbnum m the dlstrtbutton of B between P and j3, wtuch, however, does not unply equthbrmm wtth gas phase A dertvatton of tlus equatton ISgtven m Appenti B The eqns (12) and (13) with the constants detemuned by the boundary condlbons describe m full the condotions of the surface These conditions can be further related to the observed rates of production of B and C and the conversion through matenal balance equations on the reactor.
(16)
1005
EtTectof surface dflusww on the selectivity of catalyuc reactions
&** IS found, all other necessary parameters can be determmed For most prac&zai cata&ts, D is about lo-’ to lo-” cm2/sec[8, 91, h about 10V3to lo-’ cm, and the rate constants about ld to 10V2set-’ [lOj Thus the range of P - Q and M-N isfrom -10’ to 10”
Once
(17) and Flpo = F(Ai - A,,) - FC,
(18)
In pnnclple, the eqtions developed wdl descnbe the behavior of the system completely However, values must be asslgaed to a total of e-teen vanables to solve for Ao, B. and Co Fortunately, unth a few sunphfymg assumptions, the effect of flow rate, parhcle size and loadmg on the selectrvlty for C can be evaluated, w&out loss of generahty, v&h a total of e&t assumed dunenslonless vanables 3-c-m
assume that (I) bBc ) kB1#, and (u) La * kE& and khAn then the selectivity S for C, as grven by eqns (16) and (17), LS lf
4.llEsuLT~HMULATION Effectof j3owfafe
we
’
-
co -
F(Ai - Ao) = 2aTxJk
By definmg a dunenslonless flow rate .V( = 1;12aTx,&‘k&) and dlvldmg It mto the left hand side of eqn (21), we obtam
$2q_$)[g(!y)
A,-Ao
Q
- A@x*zl(X*)
x-
(g-+$jg$)
The constants &, A, and BB are related to the other parameters in eqn (19) through the boundary condltlons To calculate S, we need now only to define eight Qmensiotiess vanables, M, N, P, Q, Q, DadDaB, DBkfJD..k;“e and r*/h In the actual sunulatlon, the vanabIes P - Q (or M - N), (M - iV)/(P - Q), Q/(P Q). NW-N), n, D.,,G/D~,s, D,kf!./D,k& and r*/h are defined In this regrouped set of vanables, only N/(M - N) IS related to Ao, only P - Q and r*/lr to h, only r*/h to r*, only Q/(P - Q) and N/(M - N) to & The calculation was performed by fust combmmg eqns (14a) and (14b) to @ve
+
B@x*K,(X*)]
[(I-;)+_u*Z,(U*)] To approxunate the actual operatmg Gon&tions more closely, the curves are calculated for an arbitrary constant concentration of A III the mlet stream that corresponds to a value of lo-* for the term k/x=) (h2/D,) k&A, BY settw cK&‘,Jx,,k;b) = 10m3 arbltranly, we have
J%-&- a-40)= lo-“-
S2?rTx,,D,
10-3
>
(22)
t)lvldmg eqn (22) into eqn (21), we obtam, N-z:!
xeDa VtP) IdU*) (e&a* - QIP) - XJL VW) ZIV.W
(CB* - NIM) II
The dependence of selectivity on flow rate for a gven set of parameters 1s shown m Fig 3 ms figure IS obtamed by sunphfymg and rearrangmg eqn (16)
Q)&-[(l-$)+.JJ*Z,(U*))
9= 1U-*- 1O-3
- zl(t/(P))K,(X*)IKI(t/(P))
Zo(X*)+
Z~(t/(P))KdX*)IK,(~/(P))
Thus, with eqn (15), are then used to solve for flBB* to give B* = 8 B* 08 =
forfk= 1
= -b*V(b2--4ac) 2a
where a=H(l-a, b=Hn+(l-n)($+H$)-1 .=l-b($-H$)
-fora> +forficl
1
An umque flow rate can then be calculated for a p;lven set of dunensionless vanables Using eqns (17) and (19), (ZG,/2rTx,JL) and S can also be calculated Asslgrung dtierent values to the terms (&x=) (h’/D,) k& Ar and (xek?,Jxak8) only shifts the curves along the x axis Hnthout changmg the trend shown Ftgure 3 thus represents the observed selectlvlty as a function of flow rate for 3 different catalysts operatmg with the same inlet concentration of A As expected, at very low flow rate, or long residence tune, selectivity is h@ because the intermediate B produced has a high probability of readsorption onto j3 for further reaction At high flow rate, or short residence tune, B produced IS quickly removed from the reactor, resukmg in a low concentration of B and low selectlvlty The ddferent
I
r
.
-4
.
-2 log
3
0
I
dtifuston and thus the effect of crystalhte stze should be most nouceable F-e 5 shows that seiecavtty IS constant for small crystalhte stze (small P-Q), wbd rt drops off at large crystalhte srze Tins behavror IS due to the fact that se~ecttvny depends on the ratto of the coveraKe of B on o and fi At low P-Q, (tLn/fbB) at the mterface IS almost constant whtch results m a constant S With proporttonaUy faster mcreasmg P - Q. 8,” maeas+ than @,“, and causes S to decrease
Effect ofload& Although loadmg
2
pt 3 K&et oi now rsts on selecuv1ty curves sre calculated with tg~fi) - 10, c~tm.$) = 10-3. a = ld, (fvlr) = OlmdM-N=IO-4 values of (M - NIP - Q) correspond to Merent rattos of (&J~=) The larger ts tlus ratlo, the lower 18 the dame rate of reaction of B, and correspondmgly the lower is S Relative rates of productron of c as calculated from eqn (17) for the condutons rn Fi 3 Is proportronal to the flow rate at low dew rate, and reaches a maztmum at high flow rate ‘Plus shtft from Lear dependence to non-dependence on flow rate occurs m the regton where the selecttvrty c-s from umty to 8 constant value Frgure 4 shows the frachon of C produced from sM= ,““” equals 2uTrD.. (d6-‘?dr)[,_,* The . . Wibutmn mxeascs wdb mcreastng flow rate, and becomes the only mechamsm of transport of I3 at veryhtphfiowrate Efixt of crpJtallrtc sue The effect of crystalhte stze of a(+) was studted assummg that the reactton ISstructure msenstttve, that IS, the rate constant per sue IS mdependent of crystallite stze The calculation IS done at a constant loadmg (p/h) and constant rat108 of (N/(M- IV)), (a(P - 0)) and ((M - N)/(P - Q)) The selectrvlty ts then evaluated as a functton of P-Q (1 e h) We choose to study the condWon at very hgb flow rate when dtifuston Is the only transport mechamsm for B to get from u to /3 Under tlus condthon, selecttvtty 1s zero m the absence of
IS generally defined as the we& percent of the a @base, It IStaken here as the fractton of surface covered by Q, that IS, (r+lh)’ The calculated values of S and the relattve yrefds of C are plotted as funcuons of (f/e) m Ftgs 6 and 7, respecttvely They are calculated for the condittons of very high flow rate such that dtffuston IS the only transport mechamsm, of small P - Q such @at there IShttle crystalhte stze effect, and of constant h We have also used the same set of parameters as m Frg 6 for the points at (9/A) = 0 1, and calculated a plot of S vs +/!I by varymg R and keepmg r* constant The resultug plot IS superunposable on Ftg 6 Physically, a constant h means a constant total surface
rn
0
-4
-2
2
0 log (P-d
Cwv~s are CdFu 5 Effect of crystalhte SIZCon se!echwty culated wrth &DJ.@p) = 10, (D,&LlDAT~ = lo-‘. fl= lw, (r*/h)=Ol,(N/(M-N))=OlandQaO
I
I
I
IO
l-o-
v)
0
Oo -4
4 Contrhubon
-2_
0
log2
2
-
from dhs~on for the production of C vahles of p8rameters are the same as 111Fig 3
01
02
03
w
05
06
07
06
P Fw 6 Effect of loadm on selechvlty Curves are calculated with I&D&@,,) = 10. (D&-/D&J = lo-“, f-3= Id. P - Q = lo-‘, (IV&W-N))=01 and Q-O
Effect of surface dtffuslon on the seltcttvtty of catalytic reactions
Fii
7 Relattve yteld of C as a function of load1118 Values of parametersare the same as m FJii 6
area of catalyst, and a constant r* means a constant area of a S mcreases with decreasmg loadmg m Ftg 6 because of two effects Ftrst. as (P/h) decreases, the ratto of the surface area of P to the length of the boundary decreases Smce desorptton rs proporttonal to the surface area, whde dtffuston to the length of the boundary, the relattve mportance of dlffuston mcreases as (r*/h) decreases, and subsequently S Increases The second effect is that the relative area of @ Increase wtth decreasmg (r*/h), whtch ISeqtnvalent to mcreasmg kgBc Thrs results m a decrease m the average coverage on B, whtch enhances the drtvmg force for dtiIuston, and results m an tncrease m S Although S mcreases with decreasmg loadmg, the rate of productton of C vartes ddferently wtth loadmg Figure 7 shows the optunum loadmgs at various condutons for the rate of productron of C To the rtght of the optimum, the reactlon on B IS hnutmg and the yield decreases wrth decreasing area of B To the left of the optuuum, the reaction on a 1s lmutmg and the yteld decreases wtth decreasmg area of a
1807
catalysts with and wtthout surface dtffuston IS less stratght forward In general. tt can be concluded that under otherwise tdenttcal con&tons, surface dtiuston enhances the yteld We have assumed m the analysts that gas phase dtiuston IS mfuutely fast A more exact model would consider the htgher concentratton of the mtermedtate B over a than over @ Such concentratton gradient 1s due to a fimte reststance to gas phase transport, and its presence would only enhance the effects due to surface dtffuston On a practical catalyst, crystal&s of a of some distnbutton of sizes are randomly dtstrtbuted on the j3 surface 1113 We expect that thts would only modtfy the detads of our analysts wrthout afIectmg the general conclusions Also, modtkatton of our two henstonal model to a one dunenstonal one, such as that for stepped single crystal surfaces can easdy be made We also expect that the conclusions here can be quahtattvely extended to these smgle crystal surfaces NOt’ATlON
mlet concentratton of A Ao, Bo, Co outlet concentrations of A, B and C D surface dtiustvtty F flow rate through reactor h radius of /I regton radtus of a regton k’ rate constant per site T total number of a regtons m the reactor n equthbrmm dtstrtbutton of B between a and B region 8 fractton of surface X sue density A,
Subscnp ts
-1 desorptton process 1 adsorptton process a process on a regton B process on /3 regton
5 coNcuIsIoN
The analysts presented mdtcates that surface dtffuston can greatly afkct the observed selecttvity of a reaction The presence of surface diffuston reduces the dependence of selecttvtty on the flow rate In fact, tf surface drffuston of B 1s much faster than its desorptton, a selecttvtty of unity. mdependent of flow rate, can be obtamed Comhtrons used m ultra htgh vacuum studtes (hke molecular beam expenments) are equivalent to the extreme of very htgh llow rate This analysts shows that dependmg on the degree of surface dtffuston, results from such studies may or may not dtffer srgtuficantly from the conventional &u&es Another mteresttng result due to surface dtiuslon IS the dependence of selectwlty on the crystalhte sue No such dependence would be expected w&out surface Mumon Surface dlfEuslon also results IIIthe increase m selectlvity vvltb decreasing loadThe selectivity approaches tuuty at very low loading. as shown 111FU 6, but would approach zero 111 the absence of surface d&tston Comparison of the effect of loadmg on the yield of C on
Superscnpt * value evaluated at the boundary between a A B AB BC V
andB process mvolvmg species A process mvolvmg species B reaction A + B reactton B + C vacant site
[ll Wetsz P B and Prater C D , Adu Cut 19546 143 (2) WheelerA , Adv Cat 1951 3 258 133Thomas J M and Thomas S J , Intmductlon to Zfeterogeneous Catalysts Academtc Press, New York 1967 [4] Rys P , Act Chem Res 1976 9 345 [S] Satterlield C N and Sherwood T K , Role of Lb#wo~ WI Catalysrs Adduon-Wesley, Readmg, Massachusetts 1965 161 Shadman-Yazd~ F and Petersen E , Chcm Engng Sa 1972 27227 [73 ~JI , Memll R and Petersen E , Chem Emgng Scr 1970 [8] SchnelderP andSmlthJ.AIChEI 196814886 191 Reed E M and Butt J B , J Phys Chem 1971 75 133
HAROMI H KUNG and MAYFAIRC KUNG
1008
tutls for the species must obey the equation
PO1 Baetxold R and Somorjal G , J Cat 1976 45 94 Ill] Luss D, I Cat 1971 23 119 APPEMNXA Actw~fy on the area of surface outstde the cwcles The centers of the cucles form a net of lattice pomts of hexagonal symmetry There IS one cucle m each of the unit cells of area 4h* sm (a/3) Thus the area per umt cell not covered by that the boundar; the circle IS 4h* sin (?r/3)- ?rh2 Akmmg condltlon of (d&‘?dr) = 0 at r = h IS stdl true, eqn (17) can be modtfied to
o=PS.+&l&,-~~--BsI
We now substitute the expression c = to + RT In (8/P) mto (A2), and choose as the standard state the condltlon when the adsorbed E IS m eqmhbnum with the gas phase B, we obtam
and
- eA).e,B = (I- en - eA]mi9,B’ eg9e.B ( I- eeB9emB’
(I - P ( 1-
- ?rh‘) keBc&’ I*
(AlI
(~42)
(A3)
But for the standard state, we also have
The equation for selectlvlty (eqn 19) must also be correspondingly modified
(I-
eB - tP ), ktB,, = kS,,BB’
and Equdlbnum drstnbutlon of B at the boundary Treat the dfluslon of B as a chemical reactlon sp+(B
S).=S,+(B
S)e
where S,, S, represent vacant sites on a and /3, and (E S) represents an occupied site At eqmhbnum, the chemical poten-
(1 -e,?‘) k:,Bo= k!,#’ Substltutmg these, and an expression for emAfrom eqns (1) and (3) into (A3). we get eqn (15) The assumption of local equibrmm IS equivalent to the assumption that the chemical potentml of the moMe species (B and vacant sttes) are contmuous at the boundary