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Macroscopic clusters induced by diffusion in catalytic oxidation reactions
Shorter Commumcabons REFERJSNCES
[I] Haynes W M , Hlza hi J and Frederick N V , Rev Scl Instrum 1976 47 1237 [2] Haynes W M and Hma, M J , J Chem ?‘hemodyn 1977 9 1979 (31 Hlza M J , Haynes W M and Pamsh W R , J Chem Thetmodyn 1977 9 873 [4] Miller R C and Hlza M J , Flurd Phase EC@ 1978 2 49 [5] Haynes W M , Hlza M I and McCarty R D , Proc 6th Znt Conf on Z_.NG,Paper III-11, Vol 2 Institute of Gas Tech-
Macroscopic
chsters (Recewd
indwed
2351
nology, Clncago 1977 [6] Mollerup I and Rowhnson J S , Chem Engng Sci 1974 29 1373 [7] Tela A S and Rowhnson J S , Chem Engng Scr 1973 28 529 [S] TeJa A S , A Z Ch E J 1980 26 (in press) [9] Hlza M J , Fluid Phase Equrl 1978 2 27 [lo] Rodosevich J B and Miller R C , Advan Cryogenrc Engng 1974 19 339 [ 111 TeJa A S , Chem Engng Scl 1978 33 609
by difhdon
in a~talytic
20 aCcember 1979, accepted 4 Fdmary
Ordered state of catalysts surface layers has been recently proposed for -0~s oxidation reactions[l,2] It is known that sunultaneous occurence of a non-lmear chenucti rem9011 and ddGsion may produce penodlc structures (“dlsslpatlve struttures” by Pngozhm [3]) These phenomena have been analyzed m more detail by Barelko rt ul (see, e 8 [4]) The obmve of Uus study IS to explm the appearance of these structures for the catalyhc reachon omdation of CO on Pt
reactions
oxidation 1980)
ptvperttes of thesystem (1) Equation (2) de tcrmmes a smgle-valued dependence (provldedthatx~O,yrO,x+y~1) Mun
x(y) = b - d(b2 - kc) 2a
’
WIKXC
2k,p% - 2k_1. b = 4ktPo# - Y) + by + bco.
Q =
We consukr
the followmg readron mechamsm (see, e B [5]p 02 + 2Pt#2PtO, co + Pt#Ptco. PtO+PtCO-t2Pt+C~ co+Pto+pt+cO,
$; ii;
Assume that (I) ddfus~on of adsorbed CO molecules on the surface IS due to “humps” onto the ne~ghbounng vacanclts. (u) the temperature interval 1s such that oxygen adsorptmn IS localrfed and oxygen drffuslon on Pt surface may be negkcted Then the model which describes the process m the chem~sorbed layer wdl be as follows
c = 2k,poz(lx(y)
IS
dlfferenhable function. therefore. eqn (3) may be reduced
to oZ(Y&+Ql(Y)
y = kzpaz - k_,y - k,xy + D(zAy - yAz).
(1)
where x = [PtOl, y = CptCO].
@ d5 2+Oo(y)‘0, >
(4)
Whae
dy)=D
(
l-x(y)+yJj
ody) = kg&
dx
)
9
-X(Y) - Y) - k-z - h(Y)y
(2) At h&er denvative the coefficients m eqn (4) are not equal to zero where the fumon x(y) IS determmed It holds true for such y that x(y) G 1 Equahon (4) may be mte8rated IIYthe exphccltform By subsbtutmg
z=l-x-y=[PtJ
dY P(Y) = ;is
Using the balance conslderattons we can easdy receive eqn (1) The d&won term m eqn (I) IS not @merally accepted The &tads WIUbe presente4lm our next arhcle Consider now one-dunenslonai stabonary probkm
a2
(
al(Y) = m $*
x = 2k,pqz2 - Zk_rx” - k,xy - k,pcd.
“=s
Y)Z
we amve at g?(y) = (2A -2lf(u)
exp (2fiv)
dv) du)exp (-I,’
t(u) du), (5)
x=y=o,
whereAtsanarb&uyconstant,
le
f(Y) =s*
2ktpg(l -x - y)z - 2k-,x2 - ke
- Ucox
= 0. r(y)=% An mspectlon
of (5) lndscaw tm
+(P(Y))
(accluatetowlthm
2352 u
Y
-----
-----_
_-L-_
53
S2
--
I
I
I I 53
I
, SI
52
Y
F1g 1 non-zero multiplier) IS coincident with the rate of motion unit body with energy A in the field of potenti energy
U(y)=lf(u)exp
(21
E Fig 2
r(u)dv)du
states, Ss IS an unstable
steady
state
If in (5)
A,
A = mm{Ai,
Sl
of the
(3) If a lumped system described in terms of mechamsm (l)-(4) has three steady states (whxh are possible m a wide range of temperatures and pressures[6,7J), then the shape of the function U(y) IS similar to that shown in Fig I Here 5. S, stand for stable steady
I
---------_-__
Ad
We deal Hnth the object well-known m the theory of non-bnear waves, namely, so called sobtary wave (soliton)[S, 91 This solution corresponds to the case when at 6 -* -Cm one of the stable steady states exists on tbe surface and in a certain region the state approaches another steady state (not reaching it) (Fig 2)
CONCLUSION Existence of the above propemes of system (2), (3) indicates that at simultaneous occurence of a complex catalytic reaction and surface d&sion the appearance of ordered structures, spots, clusters. etc IS possible They may be of pure macroscopic origin The problem of stability of these formations IS beyond tbe scope of this study
Computrng Center, Stbenan Branch U S S R. Academy of Scrcnccs Krasnoyarsk 66QQ49. U S S R Instrtute of Catafysysls Slbenan Branch U S S R Academy of Sciences Novosrbrrsk 630090, V S S R
A N GORBAN V I BYKOV G S YABLONSKII
RJtFJCRENCES
111 May J A, Aduan Catal 1970 21 151 I21 Bernian A D and Krylov 0 V , Nestatswnamye I nemvnovesnye pmcessy v getemgennom katalrze, pp 102-115 Nauka, Moscow 1978 131 Glensdorf P and Pngozlun I , Temwdmamrcheskaya teonya
struktury, ustorchwostt I fluktuatsrr Ma, Moscow 1973
I I, Merzhanov A G and M Barelko V V , Kurochka Shkadmsku K G , Chem Engng SCI 1979 33 805 PI Kuchaev V L and N~latushma L M , AU-Unwn Conference
on the Mectwntsm
of Hetemgeneous-cataIytu
rcactwns,
preprint 59 Moscow 1974 161 Yabkmsku Cl S , Bykov V I, Sbnko M G and Kuxnetsov Yu I, Bkf AN SSSR 1976 229 917 AN Fl Bykov V I, Yablonsku G S and Sbnko M G , fiH
SSSR 1976 229 1356
G B , Lmear and Nonlrnear Waves Bl Wthem York 1974 [93 pismen L M , J Chem Phys 1978 69 4149
Wtiey. New
Bubble growth in P viscous Newtonian liquid (Accepted 5 March 1980) The problem of diffusion-fed gas bubble growth in a laqtudIS of interest in many areas of engmeermg The parhcular case of such growth in a highly VISCOUSthud has application in polymer foam formation One method of foam formation involves growth of bubbles by dilfusion of blowing agent from an oversaturated solution of the blowing agent in the liquid surrounding the bubbles The oversaturation may be achieved by a lowering of the system pressure, or by some other means A large body of literature exists on the subiect of phase
growth Detailed bibliographies may be found in the works of Scnvenll]. Street et al 121 and Rosner and Epstem[3] Ddfusionfed phase growth in VISCOUSliquids has been treated by Barlow and Ianglo1s[4]. Street et nl [2] and Szekely and Martms[5] Barlow and Langlo~s and Sxekely and Martins treated phase growth in Newtonian liquids wlule Street et al considered growth in an Oshvald-de-Waele power law liquid The analysis of Streef et ol is further complicated by consldenng the liquid surrounding the bubble to bc firutc, the liquid viscosity to vary
[I] Haynes W M , Hlza hi J and Frederick N V , Rev Scl Instrum 1976 47 1237 [2] Haynes W M and Hma, M J , J Chem ?‘hemodyn 1977 9 1979 (31 Hlza M J , Haynes W M and Pamsh W R , J Chem Thetmodyn 1977 9 873 [4] Miller R C and Hlza M J , Flurd Phase EC@ 1978 2 49 [5] Haynes W M , Hlza M I and McCarty R D , Proc 6th Znt Conf on Z_.NG,Paper III-11, Vol 2 Institute of Gas Tech-
Macroscopic
chsters (Recewd
indwed
2351
nology, Clncago 1977 [6] Mollerup I and Rowhnson J S , Chem Engng Sci 1974 29 1373 [7] Tela A S and Rowhnson J S , Chem Engng Scr 1973 28 529 [S] TeJa A S , A Z Ch E J 1980 26 (in press) [9] Hlza M J , Fluid Phase Equrl 1978 2 27 [lo] Rodosevich J B and Miller R C , Advan Cryogenrc Engng 1974 19 339 [ 111 TeJa A S , Chem Engng Scl 1978 33 609
by difhdon
in a~talytic
20 aCcember 1979, accepted 4 Fdmary
Ordered state of catalysts surface layers has been recently proposed for -0~s oxidation reactions[l,2] It is known that sunultaneous occurence of a non-lmear chenucti rem9011 and ddGsion may produce penodlc structures (“dlsslpatlve struttures” by Pngozhm [3]) These phenomena have been analyzed m more detail by Barelko rt ul (see, e 8 [4]) The obmve of Uus study IS to explm the appearance of these structures for the catalyhc reachon omdation of CO on Pt
reactions
oxidation 1980)
ptvperttes of thesystem (1) Equation (2) de tcrmmes a smgle-valued dependence (provldedthatx~O,yrO,x+y~1) Mun
x(y) = b - d(b2 - kc) 2a
’
WIKXC
2k,p% - 2k_1. b = 4ktPo# - Y) + by + bco.
Q =
We consukr
the followmg readron mechamsm (see, e B [5]p 02 + 2Pt#2PtO, co + Pt#Ptco. PtO+PtCO-t2Pt+C~ co+Pto+pt+cO,
$; ii;
Assume that (I) ddfus~on of adsorbed CO molecules on the surface IS due to “humps” onto the ne~ghbounng vacanclts. (u) the temperature interval 1s such that oxygen adsorptmn IS localrfed and oxygen drffuslon on Pt surface may be negkcted Then the model which describes the process m the chem~sorbed layer wdl be as follows
c = 2k,poz(lx(y)
IS
dlfferenhable function. therefore. eqn (3) may be reduced
to oZ(Y&+Ql(Y)
y = kzpaz - k_,y - k,xy + D(zAy - yAz).
(1)
where x = [PtOl, y = CptCO].
@ d5 2+Oo(y)‘0, >
(4)
Whae
dy)=D
(
l-x(y)+yJj
ody) = kg&
dx
)
9
-X(Y) - Y) - k-z - h(Y)y
(2) At h&er denvative the coefficients m eqn (4) are not equal to zero where the fumon x(y) IS determmed It holds true for such y that x(y) G 1 Equahon (4) may be mte8rated IIYthe exphccltform By subsbtutmg
z=l-x-y=[PtJ
dY P(Y) = ;is
Using the balance conslderattons we can easdy receive eqn (1) The d&won term m eqn (I) IS not @merally accepted The &tads WIUbe presente4lm our next arhcle Consider now one-dunenslonai stabonary probkm
a2
(
al(Y) = m $*
x = 2k,pqz2 - Zk_rx” - k,xy - k,pcd.
“=s
Y)Z
we amve at g?(y) = (2A -2lf(u)
exp (2fiv)
dv) du)exp (-I,’
t(u) du), (5)
x=y=o,
whereAtsanarb&uyconstant,
le
f(Y) =s*
2ktpg(l -x - y)z - 2k-,x2 - ke
- Ucox
= 0. r(y)=% An mspectlon
of (5) lndscaw tm
+(P(Y))
(accluatetowlthm
2352 u
Y
-----
-----_
_-L-_
53
S2
--
I
I
I I 53
I
, SI
52
Y
F1g 1 non-zero multiplier) IS coincident with the rate of motion unit body with energy A in the field of potenti energy
U(y)=lf(u)exp
(21
E Fig 2
r(u)dv)du
states, Ss IS an unstable
steady
state
If in (5)
A,
A = mm{Ai,
Sl
of the
(3) If a lumped system described in terms of mechamsm (l)-(4) has three steady states (whxh are possible m a wide range of temperatures and pressures[6,7J), then the shape of the function U(y) IS similar to that shown in Fig I Here 5. S, stand for stable steady
I
---------_-__
Ad
We deal Hnth the object well-known m the theory of non-bnear waves, namely, so called sobtary wave (soliton)[S, 91 This solution corresponds to the case when at 6 -* -Cm one of the stable steady states exists on tbe surface and in a certain region the state approaches another steady state (not reaching it) (Fig 2)
CONCLUSION Existence of the above propemes of system (2), (3) indicates that at simultaneous occurence of a complex catalytic reaction and surface d&sion the appearance of ordered structures, spots, clusters. etc IS possible They may be of pure macroscopic origin The problem of stability of these formations IS beyond tbe scope of this study
Computrng Center, Stbenan Branch U S S R. Academy of Scrcnccs Krasnoyarsk 66QQ49. U S S R Instrtute of Catafysysls Slbenan Branch U S S R Academy of Sciences Novosrbrrsk 630090, V S S R
A N GORBAN V I BYKOV G S YABLONSKII
RJtFJCRENCES
111 May J A, Aduan Catal 1970 21 151 I21 Bernian A D and Krylov 0 V , Nestatswnamye I nemvnovesnye pmcessy v getemgennom katalrze, pp 102-115 Nauka, Moscow 1978 131 Glensdorf P and Pngozlun I , Temwdmamrcheskaya teonya
struktury, ustorchwostt I fluktuatsrr Ma, Moscow 1973
I I, Merzhanov A G and M Barelko V V , Kurochka Shkadmsku K G , Chem Engng SCI 1979 33 805 PI Kuchaev V L and N~latushma L M , AU-Unwn Conference
on the Mectwntsm
of Hetemgeneous-cataIytu
rcactwns,
preprint 59 Moscow 1974 161 Yabkmsku Cl S , Bykov V I, Sbnko M G and Kuxnetsov Yu I, Bkf AN SSSR 1976 229 917 AN Fl Bykov V I, Yablonsku G S and Sbnko M G , fiH
SSSR 1976 229 1356
G B , Lmear and Nonlrnear Waves Bl Wthem York 1974 [93 pismen L M , J Chem Phys 1978 69 4149
Wtiey. New
Bubble growth in P viscous Newtonian liquid (Accepted 5 March 1980) The problem of diffusion-fed gas bubble growth in a laqtudIS of interest in many areas of engmeermg The parhcular case of such growth in a highly VISCOUSthud has application in polymer foam formation One method of foam formation involves growth of bubbles by dilfusion of blowing agent from an oversaturated solution of the blowing agent in the liquid surrounding the bubbles The oversaturation may be achieved by a lowering of the system pressure, or by some other means A large body of literature exists on the subiect of phase
growth Detailed bibliographies may be found in the works of Scnvenll]. Street et al 121 and Rosner and Epstem[3] Ddfusionfed phase growth in VISCOUSliquids has been treated by Barlow and Ianglo1s[4]. Street et nl [2] and Szekely and Martms[5] Barlow and Langlo~s and Sxekely and Martins treated phase growth in Newtonian liquids wlule Street et al considered growth in an Oshvald-de-Waele power law liquid The analysis of Streef et ol is further complicated by consldenng the liquid surrounding the bubble to bc firutc, the liquid viscosity to vary