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-r < 0.5, the hyperb olic funct ions are easily repre s ented by low-order p olyn"o m ials, and hyp erb olic c ollocation merges into the better known polynomial collocation method; but clearly the met hod is a valuable generalization of t he original collocation met hod , since the ap proximate s olution is b ase d on t he structure of the individual p roblem , rather t han on a fixed set of app r oximation functions . Sdlrensen and Stewart ( 1 9 8 2 ) and S.Prensen ( 1 9 8 2 ) illustrate the applica tion of hyperbolic coll ocation on s ome very c omplicated p rob lems , and the literature contains num erous examples of the app lication of high - order p oly Consequently, we s hall be content wi th two illustrations . nomial c ollocation . In the firs t illustration the one-point collocation method is discussed as a valuable link between heterogeneous m odels and homogeneous models for reactors (e . g., a stirre d - tank approximation for a pellet ) . In the second illustration collocation is applied to s olve the pellet equations in a fixed - b ed reactor m odel .
1\;
=
Numerical Methods in Reaction Engineering Example 4 One-Point collocation Consider a first-order exothermic reaction on catalyst pellets, and let the pellet model ( 6 4 ) to ( 6 8) be ap proximated by one interior interpolation point at (UloYl):
-C [y(l) 1
where 01
- y 1] =
y q, l
!3y =constant= 1
+
[Y ( J] 1 1- 0
exp
2
0(1)
( )
d y 2 du 0 u=1
+
qy ( --q0\
where q 2/( 1 - u1) Equations ( 78) and (79) with Yb El (1), y(1) in terms of Y1: =
0(1)
(77)
!3y(1)
=
=
0b
q ( l)
y
+ +
23
)
(78)
=
qEl(l)
y(l)) El
)
( 1))
( 79)
= 1
are used to express
01
and ( 80)
=
(81)
Now the collocation equation
is simplified to
( 77)
[
-C1( 1- y1) = (�*)2y 1 exp ys* 2 (� * ) R * (y )
-
1 +
:* < ; :_
y1)
]=
(82)
1
s*
=
e
�-=--
1 + q/Bi H
q / B iM
--
1 +
and
Finally, n
=
/1 0
R( )
�
R(y ) b
dxs +1
==
�*
=
1
(1 �) /2 +
B1
q,
(83)
M
1 / y exp 0
( 84)
The left half of Table 1 shows the values of the collocation constants which are needed to construct a one-point collocation solution of (s-1)/2)(u), where sis the geometry u is the zero of P1(01 Eq. (77). 1 factor. (u1,-c1 )
24
Villadsen and Mich elsen
TABLE 1
Parameters for O n e - Point C ollocation
Bi
s
u
0
1 /3
1
2
H
Bi
' Bi finite M
-c
H
-c
B '
\i
+
w
00
w
2
q
u
3
3
1/5
5/ 2
5/6
1/6
1/2
8
4
1 /3
6
3/4
1 /4
3/5
15
5
3/7
7 / 10
3 / 10
1
1
1
1
21 / 2
1
( l ( s l ) / 2) u I n t he specific ca s e Bi H BiM + oo , t h e zero of P 1 , ( ) is pr efe r re d an d n is app roxi m ated by a more accurate t wo poin t quadrat ur e for mu la : ==
-
n
=
wl
R(y ) 1 R(y
b
)
+
w2
( 8 5)
C onstants for t hi s s p ecific situation ( from Villadsen and Michelsen , 1 9 7 8 , C h ap . 5) are shown on the right h an d side of T able 1 . Equation ( 8 2 ) is an approximate s olu tion of the ge n er al catalyst pellet pro b l e m . T here is only one unknown ( y 1 ) , and the form of the equation A is the s ame as one would obtain fr o m a stirred-tank reactor m odel gr ap hic al or alge b rai c solution can be fou n d for an y v alue of t he param eters, but t he solution is r easonabl e only w hen y ( u ) can be repr es en t e d by a fair l y l o w o r der p oly n om ial N ote in Eq . ( 8 3 ) that the original S [ = c r ( - li H )D /k T r ] is modified by a fac t o r that can be very la r ge when B i H is s m all . Typical values of B i H and BiM in a fixed bed reactor ar e 2 an d 4 0 , respectively (an in dustrial S0 2 converter) and with sp h eric al catalyst p artic le s , Eq. ( 8 3 ) predic t s amplification by a factor 3 . 1 1 . T he applic a tion of Eq . ( 8 2 ) will be illustrated by t w o n um e ric al exam ples . First con si der the trivial case of a firs t - order i sot herm al reaction with BiH = BiM + oo on s phe rical pellets . The collocation equation has the e xplici t s ol ution -
.
.
-
21
and
n
=
21
7
10
21 + 2 0 ( 1 ) , the effectiveness factor is calculated by 3Bi
11 =
7M
cp
1
coth cp 1 - 1
cp 1 cot h cp 1
+
BiM - 1
( 93)
27
N umerical M e t ho ds in Reac tion E ngi neering For 0 p cot h 1 1 / cosh 4> 1
The boundary ordinate YN + 1 is e li mi n a ted an d the r e s ulti ng set o f N + 2 differential - al geb raic e q ua tion s is solved [ e. g . , by the S T I FFALG pac k a ge I n itial values of Yi • i = 1 , 2 , , N , are based on E q s . ( 3 6 ) an d ( 37 ) ] . W hen 1 0 or = 0 . YN + 3 fou n d by s ol utio n of Eq . ( 1 0 3) with Y N + 1 = YN + 2 1 2 collocation p oi nt s are used , the optimum ( xA , x B ) ( 0 . 7 5 1 4 , 0 . 3 2 3 2 ) for 4> 1 10 and a. = 1 is fo u n d wi t h four- t o fi ve - di gi t a c c u r ac y . T he advantages of the p rese nt method are easily summed up : no un necessary computation is spent in solving Eq. ( 97 ) " e x ac tly " at every in t e gr a tion step , an d all components Y 1 , y 2 , , YN + 2 • YN+ 3 are calculated with a balanced accuracy . For � 1 -+ oo at a fixed value of ex , the pellet equation cannot be solved An asymp totic solution i s , how by collocation a s i t stands i n Eq . ( 97 ) . eve r , easily derived u si ng the following transformation . Let u !f> 1 ( 1 - (;) and rewrite ( 97) as •
.
•
=
=
=
.
.
•
=
2
£....¥. d
du
- exp ( - u )
=
2
with y ( u
=
0)
=
- �� -
du
+
a
XB / (1
-
2
(1
-
x )y A
2
XA ) an d y ( u
( 1 0 7)
-+
oo
)
0,
( 1 08) u=O
The b o u n dary condition at " u -+ oo " can be applied at two different u values , say u 5 a n d 8 , and Eq . ( 1 0 7 ) is solved by collocation for each of these va l u es . If there is no apprecia ble change in dy I dul u = O , the solution i s ac ce p t ed ; otherwise, u > 8 i s tried . Alternatively , one may make a secon d transformation to obtain a finite in t e rval : =
v
=
( 1 + u)
-2
where v
=
1 at u = 0 and v = 0 at u
-+
co
( 1 0 9)
N umerical Methods
X
B
in
31
Reaction Engineering
=1 CA0
.8
.6
.4
.2
XA
= 1 - ..,.A. c
"Ao
k 1 /c A okz and of $ = FI G U RE 3 Maximum yield of B as a func tion of s 2 Lp ( k 1 / D }t . C urve s for 4> = 0 , 1 , an d oo ar e shown , and poin t s with s 2 0 , 1 , . . . , 10 are con nected . C om p ar e with Fi g . 7 . 7 of Leven spiel ( 1 9 7 2) for consecuti ve first - order reac tion s , $ 0. =
=
The tran sformation ( 109) is well c hosen in the present ex a mple sinc e fro m Eq . (10 7 ) it is e asily see n that y "' u - 2 fo r lar ge u . Fi gure 3 sho w s the boundary c urves $ 1 = 0 an d 4> 1 + for t he o p timum ( x A , xB ) as a fun c ti o n of 1Ja.2 [ w hic h is c alle d s 2 in the classic al p aper of A dashed line indi Wheeler ( 1 95 1 ) on first -or der consecutive re actions ] . cates the locu s of the $ 1 1 op timum solution s , an d at seven different value s o f 1 /a.2 ( 1 0 , 5 , 2 , 1 , 0 . 5 , 0 . 2 , and 0 . 1) other dashe d lines connect solutions with t he same value of 1 /a2 . The solution w as found by the technique ( 10 3 } to ( 1 06 ) , usin g E q . ( 97) to obtain the collocation equations T he asy mp totic solution 1 0 . Even t h o u gh i n the present ex ample a collocation solution could be found for all $ 1 values , it is worthw hile to remember than an accurate solution o f the boundary value problem c an also be o bt ain e d by forw ard inte gration from r; = 0 or from r; = 1 . T here are positive ei genvalues in the Jacobian of f , and the solution m ay e asily dive r ge , b ut an explicit Run ge - Kutta method is j u st as good as an i m plic i t method . To fin d the solution that satisfies y ( l) xB / ( 1 - XA ) , one p roc e e d s as in E q s . ( 48 ) to ( 5 0} , int e gratin g E q . ( 97 ) to ge t her with the sen sitivity eq u ation for Zy = 'O y / a y : oo
=
=
0
0
Villadsen and Michelsen
32
d 2z
Y
2 d 1;
with z
o
2¢>�ch 1
=
Y o < z;== o >
dz
1
=
( 1 10)
- x A ) y zY o Yo
d r,:
and
I
0
=
0
Havi n g obtained the correct value of Y o for a given ( xA , xB ) , a sin gle in t e gration of E q s . ( 97 ) and ( 1 1 1 ) from � "' 0 to � "' 1 gives the sensitivity
of y with respect to
(l
From the
x
A:
and (
< x ) is found from 1
1 10 ) to ( 1 1 5 ) , one
mi ght add that for explicit inte gration of the di fferential equation ( 97) , it is computationally si mpler to fin d
YO
by the sec ant met hod .
exp lici t inte gration it m ay be pre ferable to calculate z turbation o f E q .
( 97) rather than by E q .
( 111) .
XA
Also , for an
by numeric al per-
The comp utational cost is
the s ame and the computer program is simpler . Very difficult catalyst pellet proble m s have recently been solved by in genious forw ar d inte gration technique s : nonisothermal p ellet s ) ,
K aza et al . ( 1 980 ; methan ation on S un daresan and A m undson ( 1 980 ; diffusion an d re
action in a bou n d ary layer surroundin g a c arbon p article ) , Villad sen ( 1 9 8 3 ;
and Holk an d
absorp tion followed by e xothermic reaction in a liquid film ) .
Other examp le s are given in Villadsen and Michelsen ( 1 97 8 , C h ap s .
5 and 9) .
We m ay conclude that t h e model for a single reaction on a c at aly st pelle t can be treated by a standard numerical approach :
finite - difference methods ,
collocation , or for w ard inte gration from t he center of the particle , c ase may be .
as the
E xamp le s with several indepen dent reactions will frequently
require a numeric al technique w hic h is tuned to the specific p roblem .
Ex
ample 5 has shown some o f the methods that can be used to construct an efficient numerical solution , an d t he refe rences given above illustrate other techniq ues .
As a fin al comment it should be acknowled ged that the catalyst pellet model is virtually the s ame as the model used to calculate absorption with chemic al reaction .
I n particular , t he catalyst effectivene s s factor is closely
relate d to the enhancement factor w hich is used in the film model for chemi cal ab sorption .
The design of m ulticomponent countercurrent gas - liq uid absorbers [ o r
movin g- bed reactors , as i n Peyt z et al .
( 1 98 2 ) )
presents some nasty n umeri
c al proble m s - far more difficult than those encountered in the boun dary value problem s as sociated with catalyst pelle t s .
A set of exit concentrations
in the gas phase must be gue s sed , and after in te gration of the gas - phase
and liquid - p hase mass balances t hrou gh the column , one makes a comp arison with the inlet gas composition to o b t ain an iterative c alc ulation procedure .
The differenti al equations are often very stiff , an d it requires careful program min g to achieve a st able iteration .
S tead y - S tate T ubular or Fi xed - Bed R eactor Mod e l s Detaile d discussion
of
reactor models appears i n other chapters of thi s book ;
our task is to brin g up some of the common techniques w hich are applicable
in a numerical study of t he models . For this p urpose we need only one mass balance ( 1 1 6 ) and the energy balance ( 1 1 7) . A xial diffusion terms c an b e ne glected i n t he steady - st ate model since they are sm all comp ared to the convective term s . v v
z av
ay a z
=
LD 2 r v t av
_!X .1._ ax
( b:) - � x
ax
V
av
R
( 1 16)
34
Villadsen a n d Miche lsen
v v
a e az
z
-
av
where
2
=
r v pc t ac p
X ax
(x �!)
len gt h L
x
L ( - t. H ) c
+
ive
r a di al coordinate r el at
v
av
pc T p
r
R
( 1 1 7)
r
to t he tube radius rt
flui d - phase re actant concentration and temperature relative
y,e
t o a reference state with concentration c
T
R z
� 'D PC
a
1
axial di s t an c e in r e ac tor relative to the to tal re act or
z
v
Lk
r
and temperat ure
r
==
reaction rate divided by t he re ference concentration c
==
radial velocity distribution in t he reactor tube
==
radial diffu sivity of he at and mass
r
p
I n a " o o neo u s " reactor model R is a fu nct io n of fluid - p hase properties only ; the tubular reactor with or without an inert solid p ackin g is d e s c i bed by this typ e of model . T he c at aly tic fixed- bed reactor is describe d by a " hetero geneous" model in whic h R i s a fu n c t on of particle phase temp erature and concentration , but the se extra variables do not appear explicitly , and in t he numerical treatment of a c a t aly tic fixed- bed reactor mo d e l , the pellet p roblem is treat e d separately from the fluid-phase m od el [ E qs . ( 1 16 ) an d ( 1 1 7) ] - an i d e al case for solution by p artitio n in g . T he in fluence of pellet - p hase variable s on the tot al model appe ar s only in te r m s of the effectivene s s fac tor Tl , an d R i s an implicit function of the flui d - p ha s e variabla s y 9Jld e . C alculation of Tl m ay be more or le s s com p c at e d , r an gin g from the sol u t o n o f simp le al gebraic equations for a sur face reaction on impervious pe l e t to the solution of co u p le d nonlinear boundary value p roblem s , as in e a p e s 4 and 5 e ar er in this section . I n all ci rc u m st anc es one s hould try to d uc e the complexity of t he pellet model as far as possible ( Examp le 4 sho we d us how far one can get in this re sp e c t wi thout s a ri fic i n any i mp ortant feature of the pellet model) . The fluid - phase model is itself i t e complicated , an d the in e c e of re actor m acrovariable s (e . g . , the lar ge temp e r ature gradients which o cur radially reactor ) can be studied with sufficient accuracy in the u be s of a re fo r min without too many details in the pelle t - p hase desc ription . I f the r a e of e a t o n is moderate , the radial diffusion terms are small comp ared to the axial gradients , and one m ay r e u c e Eqs . ( 1 1 6 ) and ( 1 17) to a one - dimensional model . T here are c e rt ainly situation s w here this approximation may lead to loss o f major fe atures of the model , but the potential re d c tion in computer expenditure is so s i i ic an t that we feel it n ec s s a y to comment on p rop er aver a gi n techniques for t he radial gradients before we discuss suitab le numerical techniques for solution o f he fu ll od e l .
h m ge
r
i
li
i l s xm l
c
t
t
e
r
li
re
g
qu g
flu n
r ci
c
d
u
gn f
g
t
m
A v e raging o f t he S teady -State Mo de l over t he C ross Section of t he T u b e
r
D
ep
t
A ssume that t he p hy sical p rop e ti e s k I PCp and are i n d en d e n of x an d i nt e gra t e over t h e c ros s s t on of t he t u b e t o obtain t w o couple d ordin ary
ec i
N umerical Methods in Reac t ion E ngineering differential equations reactor position z :
� where R dO dz
L v av
==
dz
{ 1 18 )
a n d ( 1 1 9) for t h e average s of y a n d 0 at
35
( 1 1 8)
R
2 1 J R dx , which is approximated by R ( y , O ) : 0
=
{ 8w -
2LU r v pc t av p
==
0x==1)
L ( - ll H ) c +
v
r pc T av p r
R
( 1 19)
w here O w is t he value of 0 at t h e reactor w all and Ox:: 1 is the value of 8 just inside t he w all . In the same m anner in w hich we introduced a modified pellet heat tran s fer coe fficient in E q . ( 9 1 ) we shall define a modified w all heat tran s fer coe ffic i e n t U : a e ax
- k r t
I
x== 1 =
- 8 ) U{ O w x- 1
U{ Ow
_
( 1 20)
Expre ssion ( 1 2 0 ) is inserted in E q . ( 1 1 9) an d n o w only e and y appear i n the approxim ate reactor mo del , which c an be solved by inte gration from z = O to z = l . It remain s to calculate U in terms of U an d the radial heat conductivity k. In gene r al , U is obtained from ( 1 2 1)
where the con stant a depends on the radial velocity distribution . Villadsen and Michelsen ( 1 97 8 , C hap . 6) disc uss a pert urbation tec h n iq ue by w hich a m ay be c alculate d . T he value of a will fall between 1 / 4 for a flat velocity pro file V z = Vav and 1 1 / 2 4 0. 45 8 3 for the parabolic velocity p rofile of the Typic al result s between t hese values are laminar flow tubular reactor . =
19 48
a = - =
0 . 3958
for
�(1
v
_z_
v
=
2
av
4 - X )
an almost flat profile , and
a ::;
4 9 5b 2 + 2 3 4b + 31 1 2 0 ( 3b
+
1)
2
z
v for
v
=
6( 1 -
av
) (X 3b + 1 X
2
2
+ b)
For b = 1 / 4 this profile h a s a m1m m um at x = 0 an d a m aximum at x 'V 0 . 7 , a feature that is observed in m any experimental studies on packed - bed velocity distribution s . a = 0. 3277 for b 1/4. T he res ult ( 1 1 8) obtained by strai ght forw ar d averaging o f the m as s balance can be improved b y includin g one more t e r m i n pert urbation an aly sis from R 0. For a first -order , isothermal rea c tion ( R = k c /c ) R r =
=
Villadsen and Miche lsen
36
in a tubular re a c t or with p arabolic velocity distribution , one obt ai n s st ead o f E q . ( 1 1 8 ) ,
,
in
( 122) where D a = kRL /vav an d the numerical c ons t an t 1 / 48 is calculated by perturbation an a l y si s as discussed in Villadsen and Michelsen ( 1 97 8 , p . 2 7 1) . This result was first given in a famou s p aper b y Sir G eoffr e y Taylor ( 1 953) . Q ualit atively s p e akin g he in te rprete d the radial diffusion term in E q . ( 1 16) where D is now a true molecular di ffu sivi ty in terms of fic ti tiou s " axial dis p er s ion term" in a one - dimension al model : ,
-
� dz
=
_:£__
k L v
y
+
av
2d y 2 dz
1 __ Pe ef
( 1 2 3)
w h e re
Pe
1 92
=
ef
=
Equation ( 1 2 3 ) m� be so l ve d usin g the simple " semi - in finite" bound ary condition ( e . g . , y = 1 at z = 0 and finite for z -+ oo ) or with the more complicated boundary condition y ( O) +
l
1 dy Pe d z z= O f e
--
--
=
1
and
dy dz
l
=
0
z= 1
I n b ot h cases , one obtain s y ( z = 1 ) = exp
[ 0 p�:J J -Da
2
+
0 (p�:J
( 1 2 4)
w hic h is t he same result as that obtained by pert urbation analysis ( 1 2 2 ) . T he final result ( 1 2 4) is well known from e l em ent ary textbooks in re action en gin e e rin g [ e . g . , Levenspiel ( 1 97 2 , pp . 2 8 3 - 2 8 7 ) ] , where a " di spe r sion number" 0 v
ef = L av
1 192
=
is u sed to correct "near plug flow " data obtained in a tubular re actor . T he improvement is substantial , at least for sm all values o f t he disp e rsion Thus a t D a 1, w here Eq . ( 1 1 8) yie l d s the result y( z = 1) = number . exp ( - 1) 0 . 36 7 9 re gardles s of the contribution from t he radi al dispersion , one obtains the followin g results from Eq . ( 1 2 4) : =
=
37
N umerical M e thods in Reac tion Engineeri ng D 1 92 � v L
1
av
y by E q . ( 1 24)
y
by a " true model"
10
5
2
0 . 37 5 2
0. 3818
0 . 3892
0 . 417 9
0 . 3750
0 . 3809
0 . 3919
0 . 4038
""
( 0 . 443)
The bottom line of the table is calculated by hi gh - order collocation- as described belo w - applied to Eqs . ( 1 16 ) for a first - order isothermal reaction 2 2( 1 - x)
� az 1
y(z
=
1)
=
L av 1 1 9 2D f e
v =
a
; ax
2 4( 1 - x ) y ( z
0
=
( �) X ax
- Da y
( 12 5 )
1 , x ) x dx
The case Def -+ oo ( w hich m ay be interpreted as D "' 0 ) corresponds to a comp letely segre gated flow with no cross- sectional mixin g caused by radial gradient s .
N umerical Solution o f S teady -State Reactor Mo del
When E q s . ( 1 1 6 ) and ( 1 17 ) are discreti zed in the x direction , there appears a set of couple d ordinary differential equations (j = 1 , 2 , . . . , N) :
� = dz d6 j dz
T
M -J
a.. . C .
l. - - R ( y . , 6 . ) L
v
av
l
l
( fi T ) =
T
( 126)
m ax
( 1 2 7)
r
T he coefficients Cji are defined in E q . ( 7 0) . T he y have different values dependi n g on the method of discreti zation . Ordinary finite - difference methods , global collocation , G alerki n ' s method , or spline collocation have been used by various authors in numerous comp uter studies over the last 2 0 to 2 5 years . Equations ( 1 2 6 ) and ( 1 2 7 ) can be solved from z 0 by any of the standard p acka ge s for coupled initial value p roblems . A n imp licit or a semi -implicit method is preferable to an exp licit method because of the lar ge spread of the ei genvalue s of Q. T hi s is true in particular when ortho gonal collocation is used to discreti ze the radial derivative ; even for a relatively low order method ( N = 4 or 5) the ratio between the lar gest and the small est eigenvalue can be of t he order of 1 0 0 0 . T he structure on the right hand sides of Eqs . ( 1 2 6 ) and ( 1 2 7 ) is , howeve r , so simple that the Jacobian can be comp uted very easily . It is al so pos sible to disc reti ze E q s . ( 1 2 6 ) and ( 1 2 7) in the z direction to obtain a so -called doub le - collocation method . This method w as origi nally proposed by Villadsen and S�rensen ( 1 96 9 ) , but it w as only recently =
Vi l ladsen and Michelsen
38
elaborated into efficient computer codes by S�rensen ( 1 9 8 2 ) . T hese codes appear to be p articularly suitable w hen applied in a parameter estimation problem .
A sit uation that lends itself q uite n at urally to solution by double col location is t hat of a w all - c at aly zed c hemical reaction : the rate term appears in the boundary con dition of an otherwise linear partial differential equa tion [ Eq s . ( 1 1 6 ) and ( 1 1 7 ) without the rate terms] . Michelsen and Villadsen ( 1 98 1 ) give a det aile d discussion of this p articular type of chemical reactor model , w hich m ay be solved either by double collocation or by the S T IFFALG routine [ Eq s . ( 36) and ( 37) ] ; there is one nonlinear algebraic equation ( t he boundary con dition ) and N linear differential equations which can be trans formed into the diagonal form ( 18) and used in this form thro u ghout the calculation T he solution of E q s . ( 1 1 6 ) and ( 1 1 7) for a rate exp ression whic h is linear in y an d 9 ( or has been lineari zed from a given reference state) deserves p articular attention . C rank ( 1 95 7 ) has discussed the multitude o f practical proble m s in re action en gineerin g which are generated from t he same b asic equation , the linear " diffusion equation . " N umerically , all these different problem s are han dled by the same general techniq ue , b ut with sli ght modific ations for different boundary conditions . Let us use a linear mass balance as our " standard p roblem " : •
v( u )
�
_
az -
4 ( 1-s) /2
u
....£..._ ( au
u
( s+ 1 ) / 2
lz.) _ Da y
( 128)
au
where u W hen s = 1 an d v ( u ) i s x 2 and s = 0 , 1 , and 2 . velocity distribution , we have a steady - st ate tubular reactor When v ( u ) = 1 and z is interpreted as contact time , we have transient pellet proble m . Here we use $ 2 = kR rp 2 !D in stead vav as t he dimensionless group of parameters . Equation ( 1 2 8 ) is discreti zed by N t h - order collocation : =
the radial problem . a linear of Da = k L I R
( 129)
�
v ( llj ) , an d the ( C , b j ) are given in E q . ( 7 1) . ¥ is dia gonal with V H Di fferent side con ditions at u = U N + 1 = 1 lead to different problem modifications . _1 Diagonali z ation of M = Y ( Q * - D aD yields the standard form ( 1 8) . Michelsen and Villadseii ( 1 98 1 ) prove t hat all ei genvalues of M are real and distinct if collocation is m ade at the zeros of an d the boundary =
condition at u
( 1957) .
=
P�O , O) (u) ,
1 is any of the linear expressions discussed in C rank
T he solution appears as an N - term expansion in ei gen functions exp ( Aj z) , and t he modi fied Fourier coe fficient s are found by simple m atrix- vector m anip ulations [ see Villadsen and Michelsen ( 1 978 , C hap . 4 ) for the solution of a p articular example ] . For the case Da = 0 , Michelsen ( 1 97 9) proves t hat the ei genvalues and ei genfunctions of Eq . ( 1 2 8 ) with boundary condition
� d du
Bi
M + -2
y = constant at u
=
1
( 1 30 )
39
N umerica l Me t hods in React ion Engi neering
be derived for any value of BiM by a simple algorithm that utilizes t he eigenfunctions obtained at any specific reference value which , for example , may be either zero or infinite . This leads to substantial savings in computer time when estim atin g t he value of BiM ( and a diffusi vi t y ) from a series of measurements of y at different and x . W h en Da f. 0 , t h e diagonalization of M must , however , be performed for each value of Da. Another important speci al case , that of a finite exterior medium , i s t re ate d in Sotirchos an d Villadsen ( 1 9 8 1 ) . The boundary condition at u = 1 is given by an overall mass balance can
eigenvalues an d (BiNI >r ,
z
-
b z
q y ( )
+
l y( x , z)
s 1 dx +
0
=
a
-
(
Jz J1 0
0
R
dx
8+ 1
)
d z'
=
a -
f
( 131)
Outside the reaction medium of volume 1 (or the catalyst pellets as the case may be) there is a radially w ell mixed volume q into which the react ant m ay diffuse . The reactant concentration is Yb in the finite exterior medium at position (or at contact time t) . In Eq . ( 1 31 ) the total "mass" in the re actor and the outer medium is equated to the initial "mass" les s the total amount of reacted material. Concentrations in the two media are connected by a boundary condition of type ( 1 3 0 ) where the right - han d - side "con stant" = (BiM /2)yb ( z) . Let v( u) = 1 and write the jth collocation equation , z
a,
dy . J dz
N
I:
i=1
b.
]1
_l
c .. y.
q
1
where f is given in
f-
R(
y) ]
.
+ b.
Eq . ( 1 3 1 ) .
- b
J
1
0
For a eq
an
y dx
s+
1
N
=
L 1
�
( 1 32)
] q
j
( 2AN+1 ·) w. q
___!
+
1
J0
R dx
,1
w1. y1.
nonlinear reaction rate the N e q uations uation for t he average rate of reaction :
df dz
B\J
s+l
=
( 132)
are solved together with
N
L w.R(y ) 1 1.
for a first-order reaction the numerical approach Eq . ( 1 2 9) :
but in
( 1 3 3)
1
is j ust
like that used
Villadsen and Michelsen
40
dY =
dz
y
=
� -y
+
b*
(y 1 , Y2 ' ' ' ' , yN '
f)
T
,bN '
b*
�
(¥ =
- Da� T D aw
( 134)
0 }T
E q . ( 1 3 2 ) to ( 1 9 8 1 ) have been corrected . o ( 1 34 ) can
In
s
( 1 34) the unfortunate misp rint s in Sotirchos and Villadsen A number of important ap p li c ation s of Eq . ( 132) , ( 133) , r be easily listed : Ab sorption o f a reactant gas from a gas phase and reaction in the liquid phase [ gas flow rate va (m 3 /h) , liq uid flow rate vL( m 3 /h) , and q vL /vo l . 2 . Determination of reaction rate constant and pellet diffusivity by 1.
=
3.
concentration me asurements in a finite volume of well stirred liquid in w hich the c a taly st particles are sus pen d ed . All reactant is ori gin all y in the fluid outside the particles . Analysis of membrane reactors : a certain amount of reactant v( m 3 /h) flow s in a tub e coated with a catalyst l ayer of thickness o . R e ac t an t diffuses into the catalyst layer and reacts there . It is desired to c alculate t he len gth of t u b e required to obtain a c e rt ain average concentration in the flowin g liquid .
As a final illustration of the ge ner al numerical approach to linear steady state reactor models , one may consider Eqs . ( 128) and ( 130) with the addi tion of a s mall axial diffusion term ( 1 / Pe M ) ( d 2y / a z2) on the right - hand side of E q . ( 1 2 8 ) . T he p roble m has only ma rginal in terest in indus tri al reactor design ( radial diffusion may be accounted for by a fictitious axial dispersion term as discussed earlier , but an axial diffusion term per se is almost always insi gni fi c ant ) . A ddition of the second derivative with re spect to z in the p ar tial differential equation ( 128) does , however , lead to formidable numerical complications which have intrigued many authors [ see , e . g. , Papoutsakis et al . ( 1 980) or Michelsen and Vil lads en ( 1981) ] . It is interesting t h at wit h the exception of the inlet zone of the reactor , one may study the influence of an axial di ffusion term ( 1 /Pet\11 ) ( a 2y I a z 2 ) on the solution of E q . ( 128) , using only the eigenfunctions and eigenvalues o f the radial diffusion operator . The technique and its application to the "extended Graetz problem" discussed here and to the much more interesting asymptotic stability analysis of catalyst pellet mo dels are developed in Villadsen and Mic hels en ( 1978 , Chap . 9) . Every one of the numerical methods that have been described in this section have been based on an approximation of the x prof'J..le by a poly nomial . High -order methods such as collocation will not be able to handle th e near discontinuitie s t hat appear in t he profiles close to the reactor entrance ( t he inlet concentration m ay be 1 at all interior points an d zero at u 1 for z 0+ ) . Spline collocation with a spline point that gradually moves away from u = 1 as z incre as e s has b een successful in the computa tion of "penetration front s in the inlet zone of the re actor . One example =
=
"
N umerical Metho ds in Reaction Engineering
41
is given in Holk - Nielsen an d Villadsen ( 1 98 3 ) , and other exa mples are shown in Villadsen and Michelsen ( 1 9 7 8 , C hap . 7 ) . Another , qui te different but apparently versatile approach is to intro duce a variable trans formation n = f( z , x ) w hich as far as possible describes the combined influence of z and x on the solution near z = 0. T his is called " similarity transform ation" and i s treated in most textbooks on partial differential equations . T he reduction of the model into a problem with only one independent variable succeeds only in trivial cases , and one will usual ly en d up with two independent variables n an d z in the transformed equa tion as well as in the boundary conditions . I f , however , n has been judiciously chosen , a perturbation analysis of the transformed equation from z = 0 will give an accurate solution of the original problem in the inlet zone . T here are several examples in Villadsen and Michelsen ( 1 97 8 , C hap . 4) where this technique has been used to solve linear reactor problems , and a pellet model with a concentration dependent diffusivity is treated in the same re ference p . 3 3 9 ) . U n steady - S tate Fi xed- Bed Reac to r Mod e l s
Except for re gions of hot spots , the steady - st ate axial temperature and concentration p rofiles in a fixed bed are smooth functions of z , and the radial profiles can almost always be represented by low - order polynomials if they are not averaged aw ay as discussed in the preceedin g section . Thus ste ady- state reactor simulation is a rat her modest affair on a hi gh speed computer , requirin g at most a couple o f seconds comp utin g time to obt ain e ( x , z ) and y ( x , z ) . W hen a time derivative is included in Eq s . ( 116) and ( 1 17) there is a fundamental change in the nature of the solution . Sharp concentration and temper ature front s m ay be formed , and these front s move slowly through the reactor . To simulate the dynamic response of the reactor to a control action in the inlet ( e . g . , a jump in reactant concentration ) , one has to fol low the reactor profiles o f the dependent variables through many thousands This , of course , means t hat the amount of of fluid - p hase time constants . computation incre ases drastically , and 2 5 to 40 sec of computer time per simulation is not at all extravagant . Q uite apart from the cost of the com putation , the time it t akes to make a computer sim ulation of the unsteady state reactor makes it very difficult to use the result , for instance , in a computer control of the reactor . A compromise must be m ade bet ween the demand for accurate modelin g of the reactor and a reasonable computational effort . T he literat ure on un steady - st ate reactor sim ulation offers a bewilderin g array of model simplifi cations and numerical s hortcuts . Some of the simplifications are j ustified and should certainly be generally accepted , but others are downright silly and will lead to order-of-magnitude errors in , for example , the b reak through time of a cat alyst poison . T hus for gaseous reactants it is reasonable to ne glect accumulation terms in the reactor fluid-phase mass and energy balances and in the pellet mass balances . T he time constant for convective transport of mass through the reactor and for diffusion into t he pellet are order s - o f- m agnitude smaller than the velocity of t he reaction zone . T he pellet temperature is taken to be independent of position in the pellet ( as in Examp le 4 ) , and while the pellet time constant for ener gy transport is certainly larger than the time constant for mass transport , it is still much smaller than the thermal
42
Villadsen and
Michelsen
residence tim e for the bed , an ob servation that may justify a fur th er as sumption that Bp = e g at any time . To reduce the dimensionality o f the proble m , it is common practice to ne glect t he cros s - sectional conductive term in Eq . ( 1 1 7 ) w hen solvin g unsteady - state reactor models . This m ay be justifiable w hen the diameter -to-len gth ratio of the bed is larger than 2 0 , b ut a r adial gradient c an also be incorporated as a fictitious axial dispersion term and there is experi mental evidence that true axial conduc tive terms m ay have to be taken into account in modelin g of transient fixed bed reactor behavior . It is usually dan gerous to t amper with the rate expression to obtain comp utational advant ages . If it is reasonable to suspect a substantial re sistance to mass transfer in the pellet s , it will not do to ne glect this dis persin g effect , since breakthro u gh times that are much different from those observed experiment ally are likely to be c alc ulated . Our discussion of the pellet proble m for a sin gle reaction has also shown t hat a complicated rate expres sion is treated with almost t he sam e e ffor t as a simpler rate expres sion . T herefore , it is usually not advisable to j eopardi ze the trust wort hi ness of t he simulation b y ne glectin g the in fluence of one o r more reactants on the rate of reaction . Unstead y - state reactor models display such a variation in complexity that it b eco m es impossible to discuss s uitable solution techniques in general . At one end of t he spectrum one fin ds t he one - component isothermal gas adsorption unit w hich can be treate d analytically ( Aris and A m undson , 1 97 3 ) . At the other en d of t he spectrum there are studies of reactor con trol with several independent variables , steep temp erature front s , and with time - chan gin g kinetic and transport p arameters . Simulation of reactors for reduction o f mineral ore and simulation of explosion fronts are other com plex problems . An example that illustrates major feat ure s of unstead y - s t ate reactor behavior without bein g very complicated is discussed in Michelsen et al . ( 1 97 3 ) and Villadsen and Michelsen ( 1 978 , C hap . 9) : The reaction t akes place in t h e fluid phase , but the inert p ackin g m aterial of the reactor in troduces a l ar ge thermal residence time . A fter linearization t he model is analy zed in term s of transfer functions , u sin g collocation to account for the axial variation of the variables y and e . In the present chapter we use an example of moderate complexity to illustrate certain numerical techniques which we believe can also be applied in many other situations . Examp le 6: Simulation of co ke b urning in a fixe d - bed reac to r A re formin g cat alyst is slowly deactivated due to deposition of tarry m aterial in t he pore s . T his so -called " coke" has to be burned o ff periodic ally , and a reliab le sim ulation is important to minimize the len gth of the burn - off period without sinterin g the metal cryst allites by excessive overhe atin g of the pellets . O ri ginally , t he coke is homogeneously distributed on t he particles wi th concentration c00 • Nitro gen co n t aini n g a small proportion of oxy gen ( concentration Coi ) is fed to the reactor inlet at temperat ure T gi . A reaction zone develops and moves slowly throu gh t he reactor . When the reaction zone passes out of t he reactor the b urn is complete , an d it is succes s ful if only a sm all residual coke concent ration cc ( z) is left behind and if the temperature at no time has exceeded T max anyw here in the re actor . Durin g the operation of t he reactor a temperature wave p asses through t he b e d . I t m ay move faster or slower than t he reaction zone can be used to control t he relative the inlet oxygen concentration c oi
43
N umerical M e t ho d s in Reac tio n Engineeri n g
Both front s move exceedingly slowly comp ared velocities are 0 . 1 m / h fo r the r e ctio n zone and 0 . 9 m /h for the temperature front . A model with three equations [ ( 1 3 5 ) to ( 1 3 7 ) ] for the pellet p hase and two [ ( 1 3 8) and ( 1 3 9) ] for the phase w as proposed by Liaw et al . ( 1982) .
movement of t he t w o front s .
to the gas residence time in t he reactor .
a
Typical
fluid
y
c
c
-
=
c
c
co
=
1 at t
=
0
( 135)
( 1 36 )
1
t
ae ----2. ax
H
=
H <e - e ) + a Da n y p g p g
- .-£. p T . T
e
gl
e
T
g
=
_g T gi
( 1 37)
c
-Da ny ae
�z
g
y
g
..EK = c .
( 1 3 8)
01
H (9 - e g) p p
=
a
( 1 3 9)
The many in E q s . ( 1 3 5) to ( 1 39) ( t R c h r act eri s t ic reaction time , t H = therm al residence time , Hp = fluid -to-pellet heat tran s fer units , · T his is no more diffic ult t han the p rocedure ( 4 6 ) when t he method is explicit . For a semi -imp licit method where all ( m + l) M e quation s have to be solved sim ultaneously , it is hi ghly pro fit able to modi fy t he LU decomposition routine to utili ze the (M x M ) block diagonal structure of the [ M ( m + 1 ) ] - dimen sional J acob ian matrix . Collocation applied to p arameter estimation in di ffe rential equations was first proposed by Van den B o sc h an d H ellinckx ( 1 97 4 ) . As explained above , it m ay be an excellent app roac h to transform each differential equa tion into N collocation equation s , but u n fortun ately a n umber o f misconcep tions about the proper use of the method are app arent from the literat ure , the most misleadin g bei n g that " the experimental points should be chosen at value s o f the indepen dent v ariable ( axi al position , etc . ) whic h are zeros of a specific J acobi polynomial . " Obviously , the N collocation points s hould be chosen to m ake t he ap p ro xim ate solution l.sc a s close to :l. s as po ssible . T he collocation order N in one particular coordinate m ay have to be m uch hi gher than the available number of data points n in this direction - a situation t hat occurs i f Y s i s not a smoot h function o f t hi s particular in dependent variable . A fte r havin g solved t he model by a sufficiently hi gh order collocation method , one m ay ( without los s of acc urac y ) interpolate the solution to t he values of the independent variable w here the experi ment al value s X.e are available . I f , on t he other hand , there are m any more e xperiment s t han required for comp arison with a collocation solution of sufficiently hi gh order N ( t his may be t he case if t he experimental results are recorded autom atically ) , one applies N t h - order polynomial least - sq uare s fittin g o f the ori ginal data to obtain a synt hetic data base :X.ec at the same values o f t he independent variable where the collocation solution is constructed . B aden and Villadsen ( 1 98 2 ) give a c ritic al review of collocatio n - based T hey emp hasi ze t he i mportance of choosin g parameter estimation met hods . .
•
.
N umerical Me t ho ds in Reactio n Engineeri n g a co llocation met hod that give s t he b e s t po s sible fit between t he solution to t he ap p ro ximate model an d t he ( u n know n ) solution � s of the exact l.sc model . T heir model B is t he same as t hat proposed above : solve t he collocation e qu a t ion s , fi n d the gr adien t dr_ sc / de_ an d up d ate 2.· One m ay
51
,
either choose to solve t he collocation equations to m ac hine accuracy be fore
calculatin g l'l 2. or take j u s t on e step in t h e iteration towar d t he solution o f
t h e collocation equations be fore u p d at in g
method A ) eters .
only
E.·
A p articular variant ( their
m ay b e valu a b le to find good startin g value s
E_
for the p aram
If the differenti al eq u at ion model is line ar in t he parameters ( or
sli ghtly no n lin e ar ) , a sit uation t hat fre q u e n tl y occurs , one obtains a
very good e s timate for
£. b y
( 1 46 )
ON is t he re sidual of the differential equations when the dependent v ari ab le
l. is
rep l ac e d by �e c , t he " synt hetic" data b ase re ferred to the c ollocation points as described above , '!F 1 is an e stimate for the di spe r si on m atrix for
the re sid u al O N .
I t is only a weak function of p and may be evaluated at
any re aso n abl e point E. o .
( 147)
T h e g ene r al co llocation method m ay ,
however , b e ap p li e d for models that are nonlinear· in t he parameters as well as i n the state variab le �· I t is equally ap p li cab l e for p arameter estim ation in n onlinear coupled ordinary differen tial - eq u ation mode l s ( w here it competes wit h t h e e x plici t or semi
implicit initial value tec hniq ue s ) an d for p arameter estimation in boundary value pr o b l em s .
The best doc u m e nt e d general code for p a r amet e r e st i m at io n u sin g col
location i s developed b y S �rensen a n d S tewart .
in app en di x e s to SISren sen ( 1 982).
T h e soft w are i s shown
Special attention should be given to t he linear bo un dary value proble m s
which ar e the b a sis for Fourier expan sion o f lin e ar PD E [ e . g . , variant s o f Eq . ( 128) ] , since the se models occ u r so freq uently .
Michelsen ( 1 97 9) convincin gly demonstrated that the transport p ar am
eters ( radi al d isper si on and a wall transfer num be r )
can be e stimated with o n ly one di ago n ali z at ion o f the collocation matrix ( for B iM -+- oo ) and sen sitivi ties calculat e d analytically
from the e i ge nvalue s an d ei genvectors o f
C.
T he resultin g reduction o f computer time comp ared t o p re vious c alc ul a
one
W hen v z ( x ) = 1 , a situation that c ertainly occurs in pellet problems , m ay find the radial transport p arameter , t he wall transport parameter ,
tion s
on t he same dat a bas e w as at le ast by a factor of 2 0 .
and Da simultaneo u sly .
dure
of Michelsen
l_( T , D a )
=
T here are only trival modi fications in the proce
( 1 979) :
The solution with Da
exp ( - Da T ) �( T , D a
=
0)
f. 0
is si m p ly
( 1 4 8)
52
Villadsen a n d Michelsen
where dx_/d T = < Q - D a !) x_ an d x_( T , Da = 0) is found by the procedure ( 1 2 8 ) and ( 1 2 9) with D a = 0 . W hen V z ( x ) depends on x , a new diagonali zation i s required for each
value of Da but the sensitivity wit h respect to Da can still be de ri ve d C onsider the system quite inexpensively . 1
d dT ( x_)
Dap
\¥'X,
=
with solution
�
exp ( �
T )�
-1
x_ 0
( 1 49)
The sen sitivity zo a with respect to Da is found by solution o f the inhomo geneous linear differential equation d (! ) oa dT
=
Mz
= -D a
- V - 1v =
Jl...
w here z 0
- a
( T = 0)
=
0
( 150)
with solution
�
- ex p ( � h )
Da
J
T
0
1 exp ( - � t ) ;¥ - _l( t ) dt =
w here
s.. 1]
and
=
{ ()..�
Tr . 11
for i 4: j
r . . is an element of � = 1]
( 1 51)
exp O.. T ) 1
for i = j
� - 1;¥ - 1�
When the rate term is nonlinear i n x_ s , one p ro c ee ds more o r le s s a s described abo ve for the linear partial di fferential equation . Collocation i s used t o discreti ze t h e P D E i n the x direction b ut now t he collocation equa tions are in t e gr ated in the T di rec tio n , either by collocation as proposed by Sorensen ( 1 98 2 ) or by a step wise p rocedure . Sen sitivities can be c alculate d by numerical pert urb ation , but at least 1
for t he case of a sin gle reaction it is also pos sible to use a semi -implicit Ru n ge - K utta routine to inte gr a te the N s tate equations and the N x · m analy ti c ally derived sensitivity e quations . T he calculation of the two highet i n E q . ( 4 7 ) is not difficult since , in contrast to d eri v ati ve s fyy and fp y the case of M coupled mdependent reactions , the collocation differential equations have a very simp le structure . T he nonlinearities are found only in the m ain di a go n al of the Jacobian ;[ , w he r e as all the other ele m ent s in l are constant s . It is n ear l y impossible to give a balanced picture of the subject of parameter e st i mation within the framework o f one section of one chapter in a boo k . So many things are involved , and most o f t hem have more to do with common sense en gineerin g that with mathe m atics : the p roper choice
N umerical Metho ds in Reactio n E n gi neering
53
of experimental equipment an d analytical method , p lannin g of experiment s , and a gener al pessimism concerning the influence of all the variab les w hich for one re ason or another were not included in the investi gation .
T he methods t hat we have c hosen to discuss have been used by us and by others in many different situations , but still m any topics have been left out . It is hoped that E xample s 7 and 8 , which m ark the end of our review , will also emphasi ze some of the facets of p arameter estimation that could not be adequately covered in the p recedin g text . sion
Examp le 7 : Determination of a firs t -order ra te cons tant and a disper coefficient fro m exp eriments in a laminar flow t ub ular reactor Consider
the mass balance ( 1 2 5 ) with zero mass flux at t he wall , and consider a series of experiments in which t he velocity - average d outlet concentration y in Eq . ( 1 2 5 ) has been measured at di fferent axial positions z iri a reactor tube of given diameter . It is desired to estimate the rate constant kR and Experimental result s of t hi s kind were obtained the radi al diffusivity D . b y C leland and Wilhelm ( 1 9 56) , w ho determined D assumin g that kR was
known . T h e experimental setup is also described by Seinfeld ( 1 96 9 ) , Sein feld and Lapidus ( 1 97 4 ) , and Van den B osch and Hellinckx ( 1 9 7 4 ) , w here kR was to be determine d w hile the dispersion n umber w as assumed to be known from independent measurement s . We shall consider the possibility of simultaneous estimation of the two parameters . Write E q . ( 1 2 5 ) as
( 152) where 1/J gi ve
2 D /k R r . t
=
N
Y
=
L
The solution of Eq . ( 1 5 2 ) by collocation is averaged to
c. exp ( >. . Da z )
1
( 1 53)
1
1
w here the con stants Cj an d >. i are functions of 1jl only . A set of equivalent e xp eriments would comprise variation of va v at fixed tot al reactor len gth and diameter . Let Dar = k R L /vav r and , Da z
=
r
Da z'
where z '
z
v
av , r v av
---
( 154)
1 , the value of z' = vav r /v av . Numeric al values of Ci and of A i are c alc'ulated for 1jl = 0 . 1 , the value used by Seinfeld ( 1 96 9 ) and Lapidus and S ein feld ( 1 97 4 ) :
and the outlet
1
2 3
>. . 1
,
z=
1
c.
- 0 . 842 9
0 . 92 1 1
- 2 . 30
0. 069
- 5 . 26
0 . 007
54
Vi lladsen and Michelsen
A perturbation analysis for
l ar ge 1)1 [ see E q . ( 1 2 2 )
for the first term ]
yields A.
c
1
1 -1 + - 4 8 1j1
=
1
1 -
=
1 7681j1
2
1 1 9201)!
2
+ 0 ( 1)1 - 3 )
( 155)
3 O ( lji- )
+
2 while A. i 1\, O ( lji- 1 ) and � 1\, O ( f > for i > 1 . W hen ljJ is large enou gh to allow terms in ljJ- 2 to be neglected ( ljJ 0. 1 is barely large enou gh to permit this approximation ) , the expansion ( 1 5 3) can be truncated after the first t e r m : =
Y�
exp
[ ( Da
r
-1
+
�
4 1)1
) ]
( 156)
z'
and measurement s of y alone are not sufficient t o determine sep arate values of D� and lji. Only the product Dar ( - 1 + 1 / 4 8 \jJ) will be found by plotti n g In y versus z' We have c o n s t ruc ted the followin g " mea s u r e me n t s " of y for Dar = 2 and ljJ = 0 . 1 by high - order collocation of E q . ( 1 5 2 ) and truncatin g t he results after the third digit : •
z'
y
- In y
- In y/ z'
0. 1
0 . 826
0 . 1 912
1. 912
0. 2
0 . 687
0 . 3752
1 . 877
0. 3
0 . 555
0 . 5551
0. 4
0 . 48 1
0 . 7319
1 . 850 1 . 830
0. 5
0 . 404
0 . 90 6 3
1 . 813
0. 6
0 . 340
1 . 079
1 . 798
0. 7
0 . 286
1 . 252
1. 788
0. 8
0 . 2 41
1 . 42 3
1 . 779
0. 9
0. 203
1 . 5 95
1 . 772
1
0 . 172
1. 7 6 0
1 . 760
A c c o r di n g t o E q . ( 1 5 6 ) , t h e ratio ln y / z ' sh o u l d b e constant , an d since this is not quite so , we m ay be able to determine both parameters D ar and
lji.
Linear regres sion of t he data with z ' > 0 . 5 yields In y - 0 . 0 5 3 - 1 . 7 1 z' now , from Eq . ( 1 5 3 ) , truncate d after the first term ( w hich is certainly adequate fo r lar ge z ' , say z' > 0 . 5 ) , one obtains i ni tial estimates for Dar =
and
an d 1)!:
"Y �
(
l -
1
-
7 68lji
2
)
( 157)
55
N umerical Methods in Reac t ion Engine e ri n g
The re s ul t i s 2. o
( DRr , lji) o = ( 1 . 9 3 , 0 . 1 5 7 } . These values are used as initial values in a comp le t e least - squares pro ce dure b ased on Eq . ( 1 5 2) , which i� solved by t hre e - p oin t collocation , and on all the " experimental" p oin t s ( z' , y ) . T he sen sitivity with re sp e c t to DRr is found by differentiation of Eq . =
( 1 52) : N
>. . z ' c . 1 1
L:
d dD a
1
1'
1
exp ( >..Da
r
( 158)
z')
while the sen si ti vi ty with respect t o 1jJ i s found by numerical p e rt urbation ( although an a ly tical differentiation could also be used ) . C on ver gence is ob t ai n e d in a few iteration s , and total c om p uti n g time ( WATFlV compiler , H ar w e ll VB0 1AD p a rame t e r estim ation pro gram ) is 0 . 2 s.
The results are 1j!
1 . 994 ± 0 . 1 %
=
( 1 59 )
0 . 104 ± 2 %
1j! is su b s t an tial , con sidering that the data are accurate to di gits , an accuracy that is n o t likely to be found in practice . T he rate constant is , however , determined w i t h t he expected small s t an d ard deviation . The two parameters are s tro n gly correlated ( correlation coe f ficient 0 . 95) and t hi s , combined with the large standard deviation of lji , show s that t he use o f l a mi n ar flow reactor data for y are u n suit able for determination of lj! , b a s e d on a " known " value of D a . A small error in the rate constant will be stron gly m a gni fie d when 1/J is e sti mate d . Converse ly , an approXimate value for 1j! is all we need to determine a very good The
error o f
three
estimate
for t he rate constant , us in g exactly the same dat a . Finally , it sho ul d be m e ntio n e d that concentration measurements at t he e xi t of t h e tube , but at the centerline ( i . e . , m e as u r e m e n t s on samples
taken through a small-bore tube which is centrally placed in the reactor tube) will be much better suited for determination of the two p arameters . We have repeated t he estimation , but based on 10 " measurements" taken at x = 0 and with the same " ex p e rime n t al error . " Now 1jJ is found to 0 . 1 0 04 wi th a standard d eviat ion o f on ly 0 . 4% , while the value of D a is 1 . 997 and has the s a m e standard deviation ( 0 . 4 %) as be fore . The result , which is o bt ai n e d by a si m p le n u m e ric al analy sis of t he model from whi c h the parameters are to be extracted , illustrate s how an experimental program can be gui d e d by si m ul at ion studies .
Examp le 8: Kinetics of a homogeneous liq u i d -p h a se subs t i tutio n reactio n ( " the Dow C hemical Company tes t p ro blem " ) In 1 9 8 1 , B lau et al . from the Dow C he mi c al Company p ub li s he d the formulation of a multi-response parameter estimation problem . T he problem is co n c er n e d with an i nd u s t ri al reaction , " hidden" behind the followin g symbolic nomenclature :
HA + 2BM
=
It is a liquid - p hase re ac tion
M- an d Q+ .
( 160)
AB + HMBM c at al y z e d b y Q M ,
which i s fully
di s sociated to
56
Villadsen
and
Mic he lsen
T he problem w as given t o a number o f research group s in t he U nited State s and in E urope w ho were s upposed to cont ri bute with a solution . A com pari so n of t he solutions would , it w as hop e d , give a review of the state of the art in p ar ameter fit tin g for r eaction en gi n eerin g m o dels H ere w e sum m ari z e t he more ge n e ral conclu sions from the solution that w as sub mitt e d by o ur group ( S arup , 1 98 2 ) . T he postulated mechanism is ( B lau et al . , 1 98 1 ) : .
K
HA
2
-- A- +
H
+
HABM ( - H )
MBM
-
+
H
+
+
HMBM
T able 3 summari zes t he T he reaction is carried out in a b at c h re actor . co nc en t r ation - t i m e p r ofi le s for one of the t hree temperatures used in t he e x p er i me n tal w or k T here ar e three rate co nst an t s , k 1 , k _ 1 , a n d k2 , an d there are t hree equilibrium constant s , K 1 , K 2 , and K 3 . A ctivation energy and frequency fac t or for t he t h re e rate constants and value s o f t he three t e m p e r at ure i n depe n den t e quilibri u m constants are d esired . T he 10 s p ecies HA , B M , HABM , AB , HMBM , M-, A-, H+ , ABM , and MBM- are given by mass balances six ordinary differential e q uation s coupled w it h three eq uilibrium relation s and one condition that expresses electroneutrality . At t = 0 o n l y HA and B M are present with a s m all amount of c atalyst QM . - 17 I nitial estim ates for all p arameters are given by B lau et al . ( 1 9 8 1 ) . From the set of initial e st i m at es we shall mention only t h at K = K = 1 0 3 1 and K 2 10- 1 1 Our solution of t he problem p roc eeds in t he fo llowin g t hree step s : 1 . A na lysis of t he o ri gi nal dat a . A closer study of the dat a in T able 2 reveals an al ar min g fit o f the mass balan c e for B . Except for obvious misprints it is seen t hat .
-
,
=
c
BM
=
c
BM
-
c
H AB M
- 2c
AB
( 161)
T he data were an aly ze d by t h e method of B o x ( 1 9 7 3) an d i t w as found that t here is not o n ly one , but two linear relations between t he " ori ginal" dat a . C orr e s po nd en c e with B lau gave the following s upplementary information : a. b.
Cs M was not me asured , but back - c alc ulated from t he m ass balance ( 16 1 ) . T he concentrations of HA , H ABM , an d AB were all measured , but t heir s um was normali zed to a gree with a " known" initi al concentra
tion of HA :
57
N umerical Metho ds in R eactio n E ngi neering
Concentration Versus Time Profiles for Dow C hemic al Company Test Example ( T 4 00 C )
TAB LE 3
T he
=
Concentrations ( g mol /kg)
Ti m e
BM
HA
HABM
AB
0 . 00
8 . 3 20 0
1 . 7066
0 . 0000
0 . 0000
0 . 08
8 . 3065
1 . 6 96 0
0 . 0077
0 . 0029
0. 58
8 . 2 95 4
1 . 6 82 6
0 0 2 34
0 . 0 00 6
1 . 58
8 . 2 7 30
1 . 6 5 96
0 . 0 47 0
2 . 75
8 . 2 4 37
1 . 6 3 05
0 . 0763
3. 75
8 . 2277
1 . 6143
o . 0 92 3
4 . 75
8 . 2026
1 . 5 8 92
0. 1174
5 . 75
8. 1781
1 . 567 3
0 . 1 37 1
8 . 75
8 . 126 5
1 . 5133
0 1 93 5
23. 05
8. 0 167
1. 4075
0 . 2 94 9
21 . 75
7 . 8440
1 . 2308
0 . 4760
2 8 75 .
7 . 6 97 7
1. 0931
0 . 6047
0 . 0088
46 . 2 5
7 . 3234
0 . 7268
0 . 95 30
0 . 026 8
52 2 5 .
7 . 14 95
0 . 5773
0 . 0881
0. 0412
76 . 2 5
6 6123
0 . 206 5
1 . 2 92 9
0 20 7 4
106 . 2 5
6 2309
0 . 06 5 0
1 1 94 1
0 . 4475
124. 2 5
6 . 1220
0 . 0391
1 . 1 37 0
0 . 5 3 05
1 47 . 75
6 . 0084
0 02 44
1 . 0528
0 . 6 2 94
172 . 2 5
5 9 1 93
0 . 0 1 45
0 98 3 5
0 . 7086
1 96 . 2 5
5 . 8556
0 . 0083
0 . 9326
0 . 76 5 9
2 1 9 . 75
5 . 8037
0 0 074
0 . 882 1
0 . 81 71
2 40 . 2 5
5 . 7680
0 . 0050
0 . 8 4 92
0 . 8514
274. 25
5. 7222
0 0047
0 . 8064
0 . 8 957
2 92 . 2 5
5 . 702 1
0 . 0042
0 . 7869
0. 9155
3 16 . 2 5
5 6 72 2
0 . 00 1 5
0 76 2 8
0 . 94 2 5
340 . 75
5 6 5 93
0 . 0017
0 7 495
0 . 9556
36 4 2 5
5 . 6351
0 . 7263
0 . 97 9 3
3 86 . 7 5
5 . 6176
0 . 7112
0 . 9956
412 . 2 5
5. 6131
0 . 7063
1 . 0 00 3
4 4 2 . 75
5 . 5 99 1
0 6 92 7
1 . 0 141
460 . 7 5
5 . 5 95 9
0 . 6871
1 . 0 1 95
( hrs)
.
.
.
.
.
.
.
.
.
.
0 . 0024
.
.
.
.
.
.
0 . 0 0 42
.
58
Villadsen and Mic he lsen
TAB LE 2 ( C on ti n u e d ) C onc e nt r ati on s ( g mol /kg)
T ime ( hrs )
HA
BM
HA B M
AB
4 83 . 7 5
5 . 5 90 5
0 . 6837
1 . 0229
5 07 . 2 5
5 . 5 7 . 36
0 . 6672
1 . 0 3 96
55 3 . 7 5
5 . 5568
0 . 6 4 94
1 . 0574
580. 75
5 . 5631
0 . 6467
1 . 0551
651 . 25
5 . 5472
0 . 6408
1 . 0660
673. 25
5 . 5516
0 . 6452
1 . 0616
842 . 75
5 . 5 465
0 . 6 3 97
1 . 0669
So urce :
0 . 0 0 46
B lau ( 1 9 8 1 ) .
H
C A
=
CH
A
+
C HA BM
+
c
( 162)
AB
T his normali zation is common practice i n analysis of chromatographic dat a an d t he error is not as " crude" as that involved in ( a ) . Still , when normali zed m e as u r e me n ts are used , the error structure of t he true measurements is completely distorted and one m ay u se a stand ard le ast - sq uares app ro ach rather than one o f the more sop histicated statistical criteria .
.
T he comments re garding the data base should not be co nstr ue d as a criticism of the author of t he p ro bl em A preli minary analysis of the raw data is alw ays valuable , and t he flaws in the original data m aterial are p robably typical for what can be e xpe cted in pr actice . O u r conclusion has been to delete t he concentration meas u r em e n t s o f c from t he dat a b ase since they con t ain n o independent information . BM 2 . Preliminary analysis of the model . Total m ass b alances for A , B , and M and the three equilibrium relation s can be used to eliminate 6 out o f 10 dependent variable s to give a set of three coupled di fferential equations and one algebraic equation . T he postulate d values of the dissociation const ants K 1 , K 2 , and K 3 are so small t hat the acids are almost totally undissociated . One mi ght assume that the concentration of H+ is nearly zero , an d this reduces the model to thre e differential equations in t hree components , chosen as H A , B M , and M-. C - , CABM , and c MB M - are given by A c
A
_
=
(1
+
K
3
K2
c
HABM c HA
+
Kl
c M H BM K cH A 2
)
-1 (c
Q
+
_
c
M
-
)
( 163)
59
N umerical Metho ds in R eac tion E ngineeri ng K c
HBM-
1
c
K2
=
C HMB M Ac HA
while c A B , cA B , an d c H M B M are calculated from t he total m ass balan c e s M H for A , B , and M . E xc e p t for the ass u mp tion c 8 + "' 0 , t he r e are no ap proxim ations involved re l ati ve to the original model , and we have c he c k ed that our results for the parameters are virtually unaffected by t he assump tion c H+ "' 0 . With this assumption - which appears to b e q uite realistic-it i s impos sib le to find ab so lu te values for the equilibrium constants K 1 , K 2 , and K 3 . Only their ratios K 3 / K 2 an d K 1 /K 2 can be foun d . I f these ratios are as sm all ( c a . 10 - 6 ) a s in dicate d in t he p ro b le m de scription , the s ys t e m is extremely stiff : cH A will decrease al m o s t to zero , following p seudo - ze r o - o r der kinetic s . When cH A h as b e c o me of t he order of K 1 /K 2 , t he rate e xp re s sion chan ge s t o pseudo first or d e r in cHA . All H A would essentially have to react before the " dead- end" reaction to HABM rever ses direction an d st arts to s e n d A B M- b ack into t he main reac
tion sequence .
3. Sim u lation and parame ter es timatio n . To s im ul at e the solution � of the mode l , we used the IMSL ro ut in e D GEAR , which w as modified to include algebraic ( " e x act" ) evaluation o f the sen sitivitie s with respect to t he p aram e t er s T here are in all ei g h t parameters in k 1 , k_ 1 , k 2 , K 1 /K 2 , an d K 3 /K 2 , but s en si tiviti e s wit h respect to activation ener gy Ej and fre q ue n c y fac to r ex;, of t he rate con stants lq c an easily be co m p ut e d from t he sen sitivities with re spect to ki :
.
.£.1: a
a.
=
ll.
ll. a k
·a k a
��
aE
exp
=
a
ak aE
- �
T
=
(- � ) ll. (- � ) � T
exp
ak
( 164)
ok
T
2 4 s en siti vi t y e q u at io ns With thi s simple device the o ri gi n al 3( 2 • 3 + 2 ) are reduced to 15 e q uatio n s . Also , it is important to re sc ale the p arameters since the m a gnitude of a an d E are so diffe ren t t hat it become s difficult to har mo ni ze t he sensitivi ties . Instead of <X and E , we u s e k ( T o ) an d E / 0 d e fin e d by =
k
=
a
( �)
exp -
=
[
T
k ( T ) exp 0 T
E
(1
O
T :)]
( 16 5 )
w here T i s a sui t able re fe rence temperat ure ( chosen a s t h e middle o f the o t hree te m per at u r e s used in the e xp er i me n t a l investi gation ) . W it h t he re scaled p ar amet er s a lar ge part of the inherent correlation between t he two par am e t er s involved in k is removed . In ge ner al if a new p arameter P t f( p 0 ) is introduced inste ad of p 0 , one obtains the sensitivity wit h respect to p 1 by the simple al gorithm
,
=
Vi llads en and Mic he lsen
60
( 166)
T he Harwell code VB 01AD was used for the leas t - s quares minimization . The routine is b ased on the Levenberg- M arq uardt method , and it require s an alytic derivative s of the error vector with respect to the p ar ameters . T here w as no diffic ulty in obtainin g the minimum of the sum of least square s - about 1 0 s of CPU ti me to determine all eight parameters ( w hic h corresponds to ca . 0 . 5 s per inte gration of t he model and the 1 5 sensitivity differential equation s ) .
It i s remarkable that the inte gration of t he three equations for C H A , CS M , an d eM - t o get her with all the sensitivity equations - requires only H ere it is twice t he time used for the three quation s o f the state vector . m uc h better to use exactly derived s ensi t i vit y equations than to use numeri cal perturb ation of the parameters , w hich is consider ably slower ( by a fac tor of 6) and requires a m uc h hi gher inte gration acc uracy to give meaning ful re sults . A lso , a comment on t he initial values of K 3 /K 1 an d K 1 /K 2 i s necessary . To obt ai n the best e stim ate s for these p arameters ( 1 . 43 and 0 . 0 6 7 2 , re
spectively) it w as necess ary to start with an initial estim ate of K 1 /K 2 which w as decidedly larger than the small value 1 0 - 6 su gge sted by B lau
( 1981) .
It is not difficult to se e from T able 3 t hat c H A B M starts to decrease much before H A has been consumed , and consequently that K 1 /K 2 has to be lar ger th an w - 6 . It is , however , not at all obvious that the routine K 1 /K 2 and q 3 = K 3 /K 2 if the fails to predict the correct values of q 1 ori gi n al e stimate o f the parameters is used . T he reason is that the object function is affected only by t he ratio q 3 /q 1 , not by their absolute values when they are as small as 1 0 - 6 , and it becomes impos sible for the routine to drag the parameter values all t he w ay from 10- 6 to t heir correct values 1 of about 1 o - . =
The integration routine is much faster w hen ( K 1 /K 2 , K 3 / K 2 ) are of the T he reason is the sudden increase order of 1 t han w hen they are c a . w - 6 . of stiffness that occurs when c H A "' 0 , and t he apparent reaction order o f the reaction A- + B M -+- A B M - c han ges from zero to 1 . T he problem m ay occur q uite frequently in t he app lic ation of standard " kinetic codes , " an d it m ay be illustrated by the sim p le rate exp re ssio n ( 1 6 7 ) :
with k
"'
1 an d K
"'
10
-6
( 167)
Even a good numerical inte gration routine ( e . g . , DGEAR ) will fail to notice the rapid chan ge in the structure o f the solution when yA "' 1 0 - 6 and will happily i n t e gr at e to ne gative concentrations of A . This m ay have a disas terous e ffect on the computation of other reaction species , w hich m ay be heavily influenced by the concentration level o f component A .
R E F E R E N C ES Aris , R .
T he Mathema t ical T heory of Diffusio n and R eac tion in Permeab le C a t aly s t s , C l are n do n P re s s , Oxford ( 1 97 5 ) .
N umerical Metho ds in Reac tio n Engineering
61
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2 Use of Residence- and Contact-Time
Distributions in Reactor Design REUEL S H I N N AR
N ew
The C ity Col lege of t h e City U ntversity of New York , York , N ew Y o rk
I N T RO D U C T I O N
Tracer experiments an d residenc e - time di s trib u ti on s m eas u rem en t s have been used b y reac tion en gineers for 5 0 y e a r s . They are a p ow erful tool with an i n cr e asi n g number of application s . T he fi r s t ri go ro u s p re s e n tation was given by Danckwerts ( 19 5 3) . S i n c e then a large b o dy of literature has accumulated an d the topics are discussed in several textbooks (Froment and Bischoff , 1979 ; Himmelblau and Bischoff , 1968 ; Sein fel d and L api d u s , 1 9 7 4 ; Nauman , 198 3 ; Le v e n s p iel and B i s c ho f f , 1 9 6 3 ) an d revie w articles ( S hin n a r 1 977 ; P e t ho and N oble , 1 98 2 ; N auman , 1 9 8 1 ; Weinstein and Adler , 1967) . T racer experiments and residence- time d i st ri b u ti on s are useful to re a ction en gineers in several ways . T hey p r o vi de a di a gn os tic tool for t h e detection of m aldistrib utions and the flow pattern in side a react or . T hey a re useful in experim entally measuri n g the parameters of simplified flow models and p rovi d e i d e a s an d gui d e lin e s for creatin g and testin g such models . Finally , u n d e r s t an din g the concept of residence - time distrib ution gr e at ly sim plifie s the solution of many p roblems in reactor desi gn , incl u din g the choice of reactor confi guration , op ti mi z a ti on , and scaleup , and allows one to solve t hem without complex and ti m e c on sumin g c o mp u t a tion s . One must , however , be c ar e fu l not to expect more from the t echn iq ue s than they can deliver . It is extremely rare that one can eit her construct a reactor m o d e l o r p re di c t a r e act or p erfo r m an c e b ased solely on a residence time di s t rib u tion ( S hinnar , 1 97 8 ) in a s it uation involvin g c o m p lex flo w s . Most reac tor s are heterogeneous , and the theory as p re sented in most text books is rigorously a p p li c ab l e only to ho m o ge neou s reactors . It is t herefore important to understand the a d van t a ge s as well as the li m i tati on s of the method . This writer has been in volved both in d evel opi n g som e of t he theoretic al c on ce p t s related to their u se and in their p r ac tic a l ap plic at ion s ( N aor an d Shinnar , 1 9 6 3 ; S hinnar and N aor , 1 9 67 ; K rambeck et al . , 1 969; Shinnar et al . , 1 97 2 ; N ao r et al . , 1 97 2 ; Zvirin and S hinnar , 1 9 7 5 ; Glasser et al . , 1 9 7 3 ; K rambeck et al . , 1 9 6 7 ; Silverstein and S hinnar , 1 9 7 5 ; S h in n ar et al . , 1 973) . I n the followin g sections I t ry to p re s e n t the subjec t in a way t h a t ,
-
63
64
S h innar
s e es a al pplicatuson discussing ad ta h sorand fal s e t c al work which the future u presents a personal viewpo applicability int r c c and in that sense chapter th h the re r with based on my re n original no historical mp t be m full Theng theoretical concepts for h mog n us systems are de d in the t c u the r in the measurements and their use and a li a ns in reactor mod of industrial reactors. Next, we deal with a t on he s n ofttepletih moge ous ac o s w h m ti has e tracers. Following a i ro u contact-time distributions, a e residence-time h is useful catalytic re actors. cwat summari c re zce thedeuse of residence-time co ta t me
t r ss p r ctic a i s both van ge pi t l . T hus , by necessity , I m t exclude a large amount of t e i may b ecome useful in b t has yet to p rove its in p a ti e , this ad e own experience . Al o ug I try to p rovide fere c e s to the literature , at te will a d e to give proper credits to p ap ers or to develop ment . o e eo rive fo l lo wi sec tion . Then we discuss experimental e hn iq e s for i pp c ti o eli n g and s ng p p lic a i s to t de i g o ne re ti n a n d it ul p systems and the use of m ul i t h t we n t d c e c on c p t similar to distribution t at in Finally e and n c ti di stributions in a ly t i a tor si gn ,
,
-
.
D E F I N I T I O N S O F T H E R E S I D E N C E- T I M E D I S T R I B U T I O N , AG E , A N D F U T U R E L I F ET I M E D I ST R I B U T I O N
a homogeneous rat i outlet streams are completely t the s stable and steady and that the inlet and a timer mixed. Let us now assume t a each fluid a e e that is activated when it t rs or a t ne a i gl exit from w h nn return to s i e with the ac tor. Each r e at th exit has a e Thisscalar property i , one the rfinesi thecedistribution timetimefor thatofp t As with any b ac o rti l This h less h a given t (see Fig. fr cti F(t) eis the s s rib atige as t e im orl arl F(O) aha m i ice with Res idence- T i me D i s t r i bution
e c or as shown in F g u re 1.
Consider flow i
Ass ume tha
p r ti c l is quip p e d with the reactor and stops when it exits . I t is i mp t n to d e fi s n e hic the particle c a ot pa tic l e tim t a s oc a t d it , w hich re measures the it s en in the reactor . q uan t ty is defined as e de n p article . can de this p roperty y the fr ti n of the p a c e s for which t e residence time is t an value 2) . a on defined a the residence- time distribution . [ Alternatively , one can d e fin residence - time di t u on h fraction of p articles molecules which s a residence time l r r than t , s ply 1 F ( t) . ] C e y 0 n d F ( t ) m us t b e a on o to n c ally n r a si n g function F ( oo ) = 1 . h t
·
en e
-
,
=
ro a dF(t) d
D e n s i ty Function For any p f(t)
I n le t f l ow
F I G U RE 1
b b ili ty
=
distribution , F(t) , the
derivative t a
h t is defined by
( 1)
t
R E AC TOR volu m e V
Q
e
S t ad y s ta -
t
Out
let
f l ow Q
e homogeneous reactor.
65
Residence- and Con tact- T ime Dis trib u t ions 1.0
u. z
0
1:::> IIl a:: 1(/)
0.8
0.6
F - F r a c t i on o f p a r t i c l e s w i t h re s i d e n ce t i m e in the r e a c tor less thon (t)
Cl
::!; j:::
w
w (.) z w 0 (/) w a::
0. 4
0. 2
0.0
0.0
1.0
2.0
TIME ( I )
(a) 0. 8
---
3.0
4.0
5.0
-- -- ----
0.7
0.6 0
�
)(
0.4
0.3
0. 2 0.1
0.0
IDEAL S T I R R E D TA N K 1.0
2 .0
3.0
4.0
(b) FI G U R E 2
D e finition o f resi dence - time di st rib ution .
5.0
66
Shin nar
is called the den sity fun c t ion and is often of great in te r est .
In the case where F ( t ) is a residence - time distribution , f( t ) dt corresponds to the frac tion of fluid p ar tic l es whose resi dence time is bet w een t an d t + dt . S ome typical distrib utions of F ( t ) and f( t ) are given in Fi gure 3 . The way such distribution s are measured an d comp uted will be shown in the next s ectio n Some properties of the function will be discussed next . One should note that in the chemical en gineerin g literat ure the density function f( t ) is often designated E ( t ) and is sometimes called the res idence time dis t rib u tion . [1 F( t ) ] is often d esi gnated as l ( t ) . As there is a large literature on the mathematical p rop er ti e s of re si den ce time distribution outside the chemical en gineerin g literature , the standard notation F ( t ) and f( t ) will be used in this chapter . .
-
-
M ea n , Va ria nce , a n d Momen ts of f ( t )
The expected value or mean of t he den sity function f(t ) i s defined by 1"
=
Il l
f
00
00
=
E (t)
f
=
tf( t ) dt
=
0
f0
[1
- F ( t ) ] dt
( 2)
T his is also equal to the first moment of f(t) and is a widely accep ted meas ure of the central value of a di s t ri b ution In react or systems with con stant density , it is equal to the volume or liquid hol d up of the reaction , divided by the flow rate : .
1"
=
v
( 2a )
Q
and is also known as the time c o n s t a n t of the syste m . For purposes of com parison , it i s often useful to n or mali ze the residence- time distribution s uc h Therefore , we define that the time con stan t or mean value is unity .
e
=
t
( 2b )
w here e i s a dimensionless normali zed variable . Most of the graphs in this chapter use e . Another importan t measure i s the variance o f a distrib ution , V f ( t ) , which is defined by 00
=
f
[t
0
-
2 E r < t ) l f( t ) dt
=
2 Er t
The v al ue of 10
00
( 21)
C ( t ) dt
C ( t ) dt is es timated by plotti n g C ( t ) on a lo gari t hm i c plot
and ap p ro xi m at in g the tail by an exponential function . T he ti me constan t o f t h e exponent i s then estimated from t h e tan gent o f l o g C ( t ) for the largest values of t for w hic h the mea s u remen ts are s till reliable . In ge ne r al , the author does not recom mend t he use of frequency- respon se techniques or sin usoidal in p ut s for measurin g residence- time dis tribution s . T he only reas o n they are disc ussed i s that t hey mi ght b e available from control studies an d can t herefore serve as a startin g poin t . The reason for this broad s tatement is that whenever s uch experi ments are feasible , a proper step response in bot h direction s is also feasible , is less time consum in g , and pr ovi des far b etter information . I n control desi gn the oppo si t e is true . It is always best to get a direct estimate of t he property of the func1ion re quired for design . I n chemical controller desi gn one needs to know f(jw) and it i s pre ferable to measure it direc t ly , compared to es timatin g it from a step respon se . On t he ot her hand , to evaluate the flow p a t ter n , one
Shinnar
88
needs to know the escape probabi li ty and therefore a direct estimate of
[ 1 - F ( t ) ] i s p refer ab le .
Furthermore , the accuracy and the r an ge of w [( j w ) is neede d is different in t h e two c ases . I n controller desi gn , one needs t he overall form of f(j w ) an d accurate values n ear the phase lag of - 18 00 . Acc urate estimates of f(jw) for very hi gh an d very s mall w are of little i nte re s t in con t ro lle r desi gn , but are im p or t an t in eval u atin g at which
residence - ti me distrib ution .
in p u t that reason ably app ro xi or a pulse cannot be pe rfo rm ed , and one m us t often compro mize and utili ze other inputs . A pra c tical example is s ho wn in Fi g . 1 0 . In this sett in g one cannot directly meas ure F ( t ) of f ( t ) b ut c an still m easure an esti mate of t he Laplace tran s form . If o n e has an ap p roxi m ate flow model , one can fit it direc t ly to f( s ) as s hown in Fi g . 9d . This is less accurate than es ti ma ti n g [ 1 - F ( t ) ] by a step or p ulse e x p erime n t . With the latter on e can often get a goo d es ti m ate of the tail of t he function w hich determines the moments . On t he ot he r hand , to obtain reliable estimates of the tails from f( s ) , one n eed s to have acc urate meas urements of f( s ) for values of s w he re s -+- 0 . E s ti matin g t he s e co n d derivative near s -+- 0 from experi t m e n al ly determined f ( s ) is much more difficul t , and therefore estimates of variance are more diffic ult and have to be obtained by first fittin g a two parameter model of f( s ) and then es ti m ati n g the variance from t hi s model . T hi s p ro cedu r e involves some mistakes , but if t he variance is s mall , the re sul t s are often reasonably accurate [ 0st er gaard and Michels en ( 1 9 6 9) ] . An example i s shown in Fi g . 1 0 . I f one could p lot f( s ) i n a w ay equivalent to an escape probability , i t wo u l d a l low b e tte r in sigh t in to the meanin g of any deviation s of f( s ) from t he simplified flow models u se d to fit the experimen t al res ults . However , the author i s not aware of any met hod t h at would make thi s po s sib le . I t is sometimes p os si b le to use E q . ( 16 ) , In cases w here an e x p e ri m e nt wi t h an
mates a s tep
C(s) f(s)
=
output
C ( s) . t mp u
( 16)
to filter out the effects of meas urement noise , e s p eci ally on t he tai l . It sho uld b e s tressed t h a t t his m ethod i s alway s one of l a s t resort a n d that one alway s loses si gnificant information . In many cases the m o s t important information of t he t racer e x pe ri m e n t is c o n tai n ed in the tail ( t -+- oo ) and t h er e is no way to replace gettin g an ac c u rat e measurement of i t . U s in g E q . ( 1 6 ) merely o b s c u r e s t he fac t that the me as u r em e n t noi s e prevented an acc urate measurement of the t ai l of f ( t ) . The me thod is sometimes useful w hen one knows t ha t the contrib ution of t he tail i s small or when one is interested in b y p a s s p henomena [ which dep en d on t he pr op e rt ie s of f( t ) n ea r zero ] . In all s uc h cases it is importan t to us e an adequate number of values of s in t he ran ge 0 < s < T , where T is the exp e c t ed average residence time . B ut one s hould always be aware of the s hor tc om in gs of this m et ho d .
Residence - and C o n t a c t - Time Dis trib u t ions U S E O F R ES I D E N C E- T I M E D I S T R I B U T I O N S I N R E A C TO R MO D E LI N G A N D T E ST I N G O F I N D U ST R I A L R E A C T O R S
89
Testi n g fo r M a l d i s t r i bu tion i n I n d u s t r i a l Reacto r s Residence - time dis trib ution s a re a n i m p ort an t di a gno s tic tool in e valuatin g the performance of i n d us tria l reactors as we ll as in s ettin g up si mplifie d
flow
T hey c a n be utili zed fo r m o de li n g both homogeneous and T he most i mport a nt a pp lic a tion i s p robably chec kin g for m al distrib ution in flo w s . A la r g e number of industrial reac tors a re de In the signed to a pp roxim at e either a s tir r ed tank or a plug- flow reac tor . first case good mixin g i s essential . I n the second case , s uch as a packe d b e d re ac tor , i t is very i m p o rt ant t hat t h e fee d i s w e ll distributed over the c ro ss section and t hat t her e be n o b y p as sin g due to fa ul ty in ternals or de fec tive cataly s t lo a din g . I f a reac to r does no t p erform as p re dic ted , it is importan t to find out w hether t he poor pe r forman ce is due to an in he ren t scaleup prob le m , a faulty cat aly st , or a wron g mechanical d e si gn . A trac e r experi me n t can be a va luab le tool in distin guishin g between t he se pos s ib ili ties . If there are similar reac tors in op er ation which are acces s ib le , one can mode ls .
he t eroge neous reac tors .
con duct a t ra ce r experiment on bot h r e acto r s an d compare t he res ult s .
Oth e rwise , one can c o m p a re the results of the trac er e xp erim ent to those
p redic t e d by an ideali zed mode l .
T hey key q uestion i s how one in t erp ret s any devi ation s between the re sults of t wo r eacto r s 01' bet ween t he actual per for manc e and the mode l . Here t he escape p rob ab ility o r in t en si t y func tion , A ( t ) , i s a u s e fu l tool , a s i t pr ovi de s o n e with immediate in si ght as to
what
or
t he
deviation s m ean .
comp u te A ( t ) .
One does not even always have to actually plot
Wi t h some e xp erienc e on e c an recog ni z e t he i mp o rtant
features of the e sc ape probability from a plot of ln ( 1 - F ) versus t . T her e fore , w hat is really ne eded i s a reliable estimator of ( 1 - F ) whic h ca n usu ally be obtained from tracer experi m e n t s . T h e followin g two examples will
illustrate the method .
Imperfect Mixin g and Bypassing in S tirred- Tank Reac tors There are several problems that c an occur in the scaleup of stirred- tank reactor s .
One of t he m is t h a t the mixin g is in s ufficient to en sure that the
enterin g flui d mixes s uffic ien tly r ap i d ly with the contents of the reac tor
[ (Evan gelista et al . , 1 9 6 9b ; S hinnar , 1 9 6 1 ) .
In hetero ge n e ou s stirred- tank
t r an s fe r bet ween the p h a ses , uniformity o f turb ulence , and the other complicatin g fac tor s whic h A l t h ough these p rob are di scussed in S hinnar ( 1 96 1 ) an d T army ( 1 98 3) . lems can be studied u sin g tracer experiment s , they involve methods tha t are outside t he scope of this c hapter . T here is on e p roblem that commonly occurs in the sc ale u p of a s tir red tank reactor and can be effe c ti v ely dealt with u sin g the residenc e - time distrib ution . T his is t he case of a d e si gn where t he escape probability is to remain con stan t and is discussed belo w . When try in g to analy ze w hether the e sc ap e p robability is re al ly constant , consider the type of deviations to be e xp e c ted . One would always expec t some de vi ation a s t + 0 , as there is a minimum fin ite time for t he p ar tic l e reactors there are the additional p roblem s of mass
90
Shinnar
2
�'"'\
A( 8 )
I
\, o
IY"' \
1
\
....._
r'--A �-/...I. 0
J
__
//
I ;, 18
?----------------- - - - - - -- - - -
...... ....... .... -- -c- - - - - - - - - - -...... --::-. -.::::::: .::::: .. -
e
FI G URE 1 2 Inten sity function for i m p er fe ctly mixed stirred- tank reactors : A , short delay between inlet and outlet ( normal cas e ) ; B , delay between inlet and outlet due to insufficient sti r rin g ; C , b yp as s between inlet and outlet ; D , bypass b etwe en inlet an d outlet and stagnant r e gions due to in sufficient s t irri n g .
Howeve r , this time is usually to reach the exit ( s e e Fig . 1 2 , c u rve A) . small comp are d wit h the av e ra ge re s idence tim e T , and if T is large , on e would not notice it . I f the mixin g in te n si t y is too low , this in itial deviation might be quite la r ge , e s p ecially in tall reactors with viscous fluids . In that case one would e xpec t A (t) to look like curve B i n Fi g . 1 2 . This could be re m e died by i n c rea sin g the agitator sp eed or the agitator desi gn . Another com mon type of faulty pe r form ance is shown by c urve C in Fig . 1 2 . It occ urs i f the in l et and outlet pipes are close t o each other and mixin g i n te n si ty is insufficient to distribute the incomin g fluid sufficiently fast over the total reactor . A fraction of the liquid will reach the outlet dir ec tly creatin g a bypas s . For small t on e could then expect A { t ) to be significant ly hi gher than the average expected value , as enterin g p a rtic le s have a chance to be in that bypass . I f a particle has s t aye d in the reactor lon ger than it takes to reach the exit via t he bypas s , its c hance to exit is n ow lower t han in the be ginnin g and A ( t ) will exhibit a maximum at low values of t . I f the vessel is really badly mixed , A ( t ) wi ll even have a decreasin g tail ( Fi g . 1 2 , curve D ) . A si gn i fic an t a dv an t a ge of r e p r e s e n tin g the information con tain e d in the residence- time di stribution b y plo t tin g A ( t ) is that one can plot the expected form of A ( t ) without any comp utation s , a s A ( t ) co rr e s p on ds to the p hy sical concept of the sys tem . An actual example taken from an industrial reac tor is shown in Fi g . 1 3 ( Murphree et al . , 1 964) . T he conversion in the re actor was m uch lower t han expect ed from the pilot plant res ults . A step in p ut t racer experi m en t was performed an d the result is replotted in Fi g . 1 3a , curve E ( one i mpeller) . Fi gure 1 3b gives ln ( l - F ) a n d Fi g . 1 3c the c o r re sp on di n g curve . T he escape p rob ability is similar to c urve B , in Fi g. 12. I t in dicates i mperfect mixin g . The mixer was modified an d e q uip p ed with t wo impellers and a gain tested by a t racer step inp ut . T he results
Res i de n c e - a n d Con tac t - T ime Di s t rib u tions
91
are also given in Fi g . 1 3 , c urve G , and t he es c ape probabili ty is similar to curve A , in di catin g goo d mi xin g . T he new mixer al s o increased the conver sion to that expected from t he pilot plant . T he tra c er result s in Fi g . 1 3 , illu s t r a t e another aspect of the use of re si d en ce - tim e distributions t hat is not e vide n t from t h e escape probability B oth curves refer to a reactor with t he same volume and volu as plo t t e d . me tric feed rate , and therefo re the same avera ge re si d en ce time T . O n e can in bot h cases es ti ma te T from !0
oo
(1 -
F) dt u sin g an exponential tail
and c o m pare it to i t s known value . For curve G the estimate i s correc t , whereas c urve E u n de re s tim at e s T by 30%. T he fac t that c urve E under estimates T show s that the reactor is not well p erf us e d and has a stron gly stagnant re gion w hic h does not appear in t he plot . I f one could measure F ( t ) acc urately for lon g ti me s , curve E must cross curve G for the t wo in tegrals to b e equal . The true escape probability would look like c u rve D in
Fig . 12 . I f T i s eq ual for the t w o case s , then curves of F ( t ) m u s t cross ea c h other . Therefore , in d epen d ent knowledge of T provi d e s some i m po r t an t checks on the acc urac y of a t rac e r experimen t . A fairly constan t es c a p e p ro b a bility , in depen den t of history , which is characteristic of a w e ll - stir re d tank , occ u r s in quite a num b er of other situ ations , such as solids reacted in flui d beds , an d t he test describ ed here appli e s to all these si t uati on s . Packe d -B e d a n d T ri c k l e - B e d R eac to rs
In the desi gn of p ac k e d - b e d and t ric kle - b e d reactors , it i s i mportant to en s ur e t hat the feed is we ll distributed and th a t th e cat aly s t is a rran ge d in s uc h a w ay that t here are no p r e fe r r e d p a t h w ay s wi t h lower pre ssure drop s
F
ON E IMPELLER 0
1 .0
(t l
o.s
o
0. 5
(a )
0 0
o o o
0 0
o o 0 o o o
TWO I M P E LL E R S
a a 0
0
1 .0
°
o o
1.5
o@
a@
2.0
T I ME -
FI GURE 1 3 Residence- time distribution for reactors with i m per fe c t m1xm g . ( a ) Experimental residence - time distrib ution for a s ti rr ed - t ank re actor ( from ) Woo dr o w , 1 97 8 ) . The reactor did not matc h the pi lot plant and gave low conversion . A t rac er experiment showed a s ever e maldi strib ution of the flow . A s eco n d impeller wa s install ed , whic h im p rove d the flow pattern as m e a s u r e d by a residenc e - time d is t rib ut io n a p p roac hi n g an i deally s ti rred tank . T he reactor then matched the pilo t plant . Here the i mp e r fe ct mixin g is not directly recogni zable fro m curve E . I t can only b e recogni zed by c o mp a rin g the estimate of f ( 1 - F ) wi t h the kn own value of T e s tim a t e d from volume divided by flow rate ( see the t e xt ) . T his indicates that in case E t h e reactor is not w e l l p e r fu se d . ( b ) C um ulative re sidenc e - time di s t rib u tio n . ( c ) E scape p rob abilit y (in t e n si ty func ti on ) .
92
Shinnar
Gli
B
D 0
0 D
D
D
D
0
0
0 0
0
T W O I MP E LL E R S
D
0
0
0
0
0
0
0
0
D
0
1 - F (I)
0
0 0 0
ONE IMPELLER
o
0
0
TI M E -
(b) 3.0
ON E IMPE LLER
..1\ ( t }
2 .0 1.0
(c)
0 0 0 0 0 0
0
0
o o 0 0
0 0 0 0 0
T W O I M PE LL E R S
D 0 0 0 0 D D 0 0 0 0 0 0 0 0 0 0
0. 5
FI G U RE 1 3 ( Continue d )
1.0
2 .0 1.5 TIME -
Res idence- and Contac t - Time Dis tribu tions
93
which create a bypass . B efore proceedin g with analytical examples , one must first define the concept of bypass and stagnant regions . A well- designed packed bed approaches plug flow and will lead to a A ( t) function s uch as curve B in Fi g. Sa . Most of this is due to the fact that in a randomly packed bed , the path len gth for different molecules is unequal and part of it is due to mixin g in the interstices between the par ticles . W henever f( t) is not a delta function , one can in theory represent the pathways of the individual molecules by an organ pipe model such as the one shown in Fig . 14 . As the pathways are of unequal len gth , one can conceive of the shorter one as a bypas s . In general , this is not the prime concern of the diagnos tic test . One is lookin g for m uch stron ger differences caused by maldistribution of flow . One can utilize the residence- time dis trib ution in this settin g because the typical flow distribution in a packed bed has a A function similar to that of curve B in Fi g . 5a with no peaks . If the maldistrib ution is stron g enough to cause a peak in the A function , it indicates that one i s dealing with maldistribution in flow and not simply a backmixin g p rocess . There are some exceptions to t he foregoin g s tatement . I f a tracer is adsorbed , it could lead to a peak in 1\ The same applies if there is a stron g diffusional resistance inside the catalyst particle or a mass transfer resistance to the particle . Theoretical examples of such cases are illustrated in Fi g . 1 5 , which shows two examples of a packed bed with a significant chromatographic effect . The residence- time distribution s in Fig . 1 5 are based on the standard equation used in chromatography or ion exchan ge ( S hinnar et al . , 1 97 2 ; Hiester and Vermeulen , 1 9 5 2 ) . T hese c a ses demon strate that to interpret a residence- time distrib ution properly , one needs ei t her a good understandin g of the system or a good data base , such as a tr acer experimen t from a pilot plant or a similar in dustrial reactor . Exam ples are illustrated in Fi g . 1 6 , which shows tracer studies from an indus trial tric kle · bed used for hydrotreatin g fuel oil . C u rv e A is from a pilot plant and is similar to curve B in Fi g Sa . C urve B is from an industrial reactor t hat performs well an d shows a small peak in A ( t ) . Curve C is from an industrial reactor that di d not perform well . I t shows a strong peak in A and indicates a strong bypas s . Shinnar ( 1 9 7 8 ) reports that this bypass was correctable b y a change in the design which led to better reactor performance . I f the bypass is really stron g , one would expect f( t ) to be double peaked. This occ urs rather infrequently and a stron g peak in 1\ is a much more sensitive test . For a packed bed one can also check the flow distribution from the Laplace transform . For near- plu g- flow conditions the Laplace transfer {unction should be oJ the for m exp [ - 1: i' ( s ) ] . One can check the form of P ( s ) by plottin g ln [ f( s ) versus s . For one- dimensional diffusion with very large Peclet numbers , •
.
FI G U R E 1 4
Alternative flow model for a packed bed .
94
f(s)
Shin nar
-
exp ( Pe / 2 ) [ 1
( 1 + 4 -r s / P e )
1/2
( 22)
]
Equation ( 2 2 ) is approxi mate on ly for hi gh values o f P e ( Pe > 2 0 ) . On e can ( 2 2 ) by plottin g ln [ 1 /f( s ) ] - 1 vers us -r s {ln [ 1 / f ( s ) ] } - 2
check the fit to E q .
·-
which s hould give a s t rai ght line with s lope -rs and i n te rc e p t 1 / Pe ( 0s ter gaard and Mi c he l s en , 1 9 6 9 ) . However , plottin g ln [ f ( s ) ] ve r s us s might be s u ffi ci e n t an d should also be close to a strai ght line with s l op e - 1 n e a r s + 0 . I f in this plot the slope near zero is less than -r or there are 1 / Pe] ,
s t r on g c han ges in t he slope ( see Fi g .
9 ) , it in dic ates a s tron g deviation It is extremely diffic ult to dis tin gui s h sole ly by mean s of a tracer exp eriment bet ween maldistribution s ca u s e d by a ba d di strib utor and those c aused either by di ffu sio n proces ses in side the particle or by adsorption p roc e s s e s ( see Fi gs . an d 2 6 ) . For t un ately , in an industrial case , one often has a reference case from another we ll - p er formin g reactor which can be used for compari son . One s hould , fro m a well- dis trib uted plu g - flow reac tor .
15
2 .0 ,...-----,
1 .6
f {I)
1-\ I \
I I I I
1 .2
\
I I I I I
0.8
\ \ \ \
\
I
0.4
\ \ \
).. = 1
\ ).. \
= 10
\
\
'
...... ""!- _ ��--...... 0.0 L____L____,.L_____J____-===-3.0 2.0 1 .0 0.0
(a} FI G U R E 1 5
( a ) Resi dence - time distrib ution of a chromato grap hic
( F rom S hi n n a r
f
=
et al . ,
e->-• [o(t -
( b ) C u mul a tive
-r)
1 9 7 2 ; H i e st e r and V e r me u l e n +
; e - A. ( t - -r ) / r�
t
1� ,
, 1952 . )
I � >. � l( t r
re sidenc e - time distrib ution for the column
( c ) E scape p robability
( inten si ty
in
c olu m n :
•>)]• = 0. 5 part ( a ) .
func tion ) for the colu mn in part ( a ) .
r
=
1
95
Residence- and Contact - Time Dis trib u t ions
' 1-F ( t )
'
'
'
'\
'
'
'\
\
\>.
\
=
10
\
\
\
\
(b) 6
r-------� _ .... ...... / ,..,
4
A(t)
2
I
I
I
I
I
I I
I
I
I
/�
/ =
/
/
//
10
>.
v
=
1
0 L-----L-----�--�--� 0.0
1.0
(c) FI GURE 1 5
( C ontin ued )
3.0
\
S hinnar
96 3
z 0
� 2
1-
P I LOT P LANT
:J l.L. >1-
(/) z w 1z H
INDUSTRIAL R E A CTOR ( C )
0
0
10
FI GURE 16
et al . ,
20
TIME
( min )
T racer experiments in a
30
40
trickle -bed
1 978 . )
reactor .
( From Shinnar
however , be careful when usin g pilot p lant re sults as a comparison unless
the pilot plan t has the same linear velocity .
A ch an ge in linear velocity
c han ges the m ass transfer coefficient s an d t herefore also the residence time di strib u tio n in a packed bed .
trickle bed .
It also c han ges t he flow re gi me in a Often this can be predicted by fluid dynamic condition s .
U se of RT D i n Reactor Mode l i n g
T racer experi men ts a n d resi dence - time distribution s are just one of many One seldom starts in a vacuum tools in s tudyin g flow systems in reactors . and us ually has other information about the r e actor In principle , one As F ( t ) meas ures a linear p roperty wan ts to un ders tan d its fluid dynamic s . of a n onlin e a r system , the information obtainable from it is alway s incom p lete , although it can be quite importan t . I f a reactor model is available , one c an compute the expected res ults of a tracer expe ri me n t and comp are it with the actual experimental resul t s . I f one w ants only t o con firm t he model or meas ure its parameters , doint it via a re si d ence - time distrib ution is an unneces sarily comp licated te c hnique However , the use of residence- time distrib ution can be of benefit in two important ways . Firstly , one n ever exp ec t s models to be completely accurate , so t h a t re sults which deviate sli ghtly from t he prediction s of t he model are not sur U sin g I\ ( t ) gi ve s a p hysical i n si ght a s to t h e meanin g o f the de p ri sin g . S econ d , h ( t ) gi ve s in formation in a form th at i s a t leas t partially viation s . .
.
One still has a p hysical model o f s t e a dy st a te and tracer be havior b ut does not need any a p ri o ri assumption s as the the exact flow model . I nterp retin g the res ults in such a way forces one to consider what model free .
-
Residence- and Contact-Time Dis t rib u t ions
97
other explanations or alternative flow models may be consistent wit h the observed data . T his type of approach is very important in any reactor desi gn p roblem ( Overcashier et al . , 1 959) . As a n example , one might cite some o f the early tracer experiments in fluid beds , w hic h are represented in Fig . 1 7 . A low fluid b ed has a tracer response quite similar to a stirred tank , b ut representin g it in the A ( t ) domain im mediately indicates that one i s dealin g with a bypass p roblem , which was s ubsequen tly confirmed in other ways . It is often useful to have a knowledge of the p roperties of residence time distrib utions o f different flow models . These are shown i n T able 1 . One must always remember that transport processes are nonlinear and cannot be described adequately by simplified models . F ( t ) is a linear property of such a model and the only time one can use it to p redict re actor performance is in first- order reactions in homogeneous system s , a problem that is not often encountered by the reaction en gin eer . However , simplified models are a very valuable tool . I n S hinnar ( 1 978) the term learning models was introduced to distin guish them from models used in actual desi gn predictions . T hey provide an un derstandin g of how the tran s port processes might affect the chemical reactions an d give some guidance for de si gnin g the scaleup . Fortunately , many reaction s are not very sen sitive to scaleup . N evertheless , it is important to be able to recogni ze those cases where significant scaleup problem s may be expec ted . There is one important case where models derived from t racer experiment s are di rectly useful in reactor desi gn . In many reactor p roblems , a plug - flow re actor is t he op timal configuration or if not optimal , is the only design that is safe for scaleup . As real reactors are seldom true plu g- flow reactors , one wants to know how closely the desi gn approaches plug- flow and how the deviations could affect reactor performance . Here one utilizes the fact that if deviations from plu g flow are small , one s hould get a reasonable estimate about their impact from any model that has a similar residence- time distrib ution . T his is equivalent to an asy mp totic expansion around a solu tion retainin g only the first -order term s . In such a case , one could use either a model based on one- dimen sional diffusion or a stirred tank followed by plug flow or a series of stirred tanks . The latter is preferred as it is easier to compute , and t he additional complexity of a diffusion model is not justified for cases where the real phy sical transport processes are not molecular diffusion . It is also more similar in its form to act ual measure tracer responses , as compared to a sin gle stirred tank followed by a plug flow reactor .
It is common to derive t hese simplified asymptotic models by demandin g the variance of the residence- time distribution of the model is equal to that of the actual measured residence- time distribution . The variance is then expressed as an equivalent P eclet number by the relation Pe : 2/y 2 , where is t he coefficient of variation as defined earlier . For a series of stirred tanks the equivalent P eclet number is approximately 2n . It will be shown later that such models give similar kinetic performance as long as the deviation from plug flow is small . These rela tio n s make sense for Peclet numbers larger than 10 ( pre fe rab ly 20) . For reactors in whic h the P eclet numb er is s maller , simplified models based on one- dimensional diffusion or series of stirred tanks make no p hysic a l sense ( unless , of course , one is really dealin g with three stirred tanks hooked up in series ) . that
y2
98
Shinnar
1 .0
0. 8
0.6
�
0. 4
"
1 - F (BJ
""
"
0. 2
0.1
"
�
"
e
(a )
"
"
3
2
0.0
"
4
2.0 1 .5
A 1)
Two nonidentical ideally mixed ve ssels in serie s ( T and T 2 are the 1 parameters of the t w o ves sels )
D en si t y :
f( t )
=
'1
1
-t / T
'2
(e
1
e
-t /T
2)
S hinnar
1 00 TABLE
1
( C on tin ue d )
Cum ulative distrib ution :
1 - F ( t)
=
1 _ ;:.._ ..:;
_
1
1
Expected value :
Coefficient of variation :
2
T1 Laplace tran sform :
In ten si ty func tion :
Special values :
L( f , s )
1\ ( t )
1\ ( 0)
E(e
=
11 2
- st
T
+
) =
2 2
�---7-:-.,...-----:(1
1 1 1s ) ( 1
+
-1
+
1 2s )
-t /1 t/ 2 1 e __--,-___e'----:--
-t/1 1
=
J\ ( oo )
2T
+
-t/1
- 1 e '/,
2
0
=
min (;1 ' ) 1
=
T2
Plug - flow ve s se l Den si ty :
f( t )
C um ulative distrib utuon
1
=
unit Dirac function
F(t)
=
Expected value
f
t �
1 1
= 0
Coefficien t of variation :
y
Laplace trans form :
L ( f , s)
I ntensi ty func tion :
1\ ( t ) =
T wo
t
1 02
TABLE 1
Shinnar
( C ontinued )
L ap l ac e t ran s form :
L( f , s )
I n te n sity
A (t)
fu nc ti on :
=
E(e
=
S pecial value :
A ( O)
)
=
ITe
-t !c (1
A ( t)
- st
-
IT ) e
2
- s -r
1
+
(1
-
IT ) -:-= 1 +'--' s
1
2
+ IT
1
=
-
=
---
1
-
rr
E s ti mation of Model Pa rameters f rom R e s i dence- T i me D i st r i buti o n A s di scusse d in the p receding section , to esti mate the kinetic impact of s mall de v i ations from p l u g flow , one e stimates an equivalent P eclet number from the coefficient of variation of the residence - time distrib ution , One can direc tly use either f( t ) or F ( t ) [ s e e E qs . ( 1 ) to ( 3 ) ] . In the litera ture , re sidence - time distributions are often used to estimate simultaneously several param eters of a complex mocel . The author doe s not recommend this . Resi dence - ti me distributions a r e hard to measure accurately and are quite constrained . In a homo geneous flow , the first moment is fixed by V /Q . T he limit s of experimental accur acy seldom allow one to measure more than t wo additional momen t s . Often only one additional moment , namely the variance , is all one can reasonably expect to obtain . Therefore , one can not estimate more t han one or two param eters from such experiments . Fur thermore , the models them selves are not accurate , and estimating multiple param eters for approximate models from a si n gl e exp eriment i s a doubtful procedure at best ( Overcashier et al . , 1 9 5 9) ,
U s e of T racer E x pe ri ments i n Col d- Flow Mod e l s and D e si g n of R ea c to r I n te rna l s T racer experi ments and residence- time distrib ution s are becomin g a widely tool in this area . In principle , t here is very little difference in the approach from t hat w hich was disc ussed in the p reviou s sections . One w an t s to make sure that one of the limitin g models , such as a stirred tank or plug flow , is app roxi mated as closely as possibl e . Q uite often one uses multiple tracers ( which will be discussed later ) , b ut the p rinciples remain the same . The followin g two exam ples may help illu strate this point , Baf fles , especially horizontal baffles , allow one to modify the complex flow in a fluid bed such that it approaches plug flow quite closely . Properly de signed baffl es achieve this by breakin g up the bubbles and stagin g the bed ,
used
1 03
Residence- a n d C o n tact- Time Dis trib u t io n s
reducing solid mixing bet w een top and bottom and th e reby also reducin g recirculation of gas either adsorbed on the catalyst or entrained in the dense p ha s e inside p ar ti cle clusters . An example of this effect and its me as u re m ent by a tracer expe riment is gi ven in Fig . 1 8 [ for details , see Ov er ca s hie r et al , , 1 9 59] . Fi gure 1 9 gives an example of the study of gas dis t ribu tor s in fluid beds. In Fi g . 19 some of th e f( t ) plots ar e double peaked , w hich clearly indicates a maldistribution . Here both f(t) and the e sc ape probability A ( t) clearly indicate that the desi gn of the dis t rib uto r is unsatisfactory and leads to a maldistrib ution in t h e overall reactor , w hich can be corrected by a proper design . This example ill u st r a te s the importance of a p rop e r base case in comparing al t ern ativ e s Fluid beds , especially short fluid beds , have maldistributions of flow caused by the formation of larger bubbles . These are due to inherent hy drody n ami c ins tabilities and c a nno t be pre vented by the desi gn of the distributor . In the cases shown in Fi g. 1 9 , some o f t he distributor d e si gn s give maldistributions which are much more p r on ou n ced c om pa r e d to the best achievable case , which is a p oro u s plate . R e siden ce ti me distributions cannot distinguish between t h es e two ph enomen a and should always be considered only an additional , al bei t important dia gn os tic tool . .
-
BAFFLED
UNRAF F L E D
2
(a )
(8 )
4
FIGU RE 1 8 ( a ) E xp e ri mental residence - time distribution for gas fl ow in a fluidi zed - be d reactor ( gas velocity = 1 . 6 ft / s ec ) - inten si ty function . ( From Overca shier et al . , 1 9 5 9 . ) ( b ) Cumulative residence- time distrib ution for the flow in part ( a ) .
1 . 0 ,...---..;;;;::-----,
o.e
0.6
0.4
0.2 1·F( 8 )
0. 1
0.0 8
0.0 6 0.04
0.0 2
0.01
0
2
9
( b)
3
4
( Continue d )
FI G U RE 18
f (t)
s s no z z l e p l a t e s an dw i c h g a s d i s t r i b u t o r
poro u s p l a te
c
Umf
_ u_
Umf u
=
=
1 .7
2 .0
F I G U R E 1 9 Residence- time distribution measured wi th di ffe re n t gas distribu tors in a fluid bed , as a fun c tion of fluidi zation indexes u / u ( From Woodrow , 1 9 7 8 . )
mf
"
1 05
Res idence- and Con tac t - T ime D is tri b u tions Mea s u remen t of Flow R a tes a nd R ea c tor
Volume o r Hol dup
In hydrology and physiology , tracer exp eriments are wide ly used to measure
flow rates and volumes [ Wein stein and Du dukovi c ( 197 5) ] . In reacti on en gin eerin g it is possible in most cases to measure flow rat e s directly . Excep tion s are reactors with internal circulation caused by density an d pressure differences , where t racer experiments are sometimes useful to measure flow It is not a very ac c urate method and should b e used rates u si n g E q . ( 2 ) . only as a last resort . In homogeneous reactors , the volume or h ol d up is almost always know n . One can then compare t h e volume es ti mate from t he residence- time di stribu tion with the known volume . To do thi s , one has to esti mate the contribu
tion of the tail o f either F ( t ) or f ( t ) to the first moment . If the volume estimated from the first moment i s si gnificantly smaller than the know n volume , the ves s el is not w ell p er fu sed and has stron gly stagnant region s . An example o f this was shown i n Fi g . 1 3 . In multi phase systems the rela tive volume and holdup of solid liquid and gas are sometimes hard to measure , and he re tracer experiments can be valuable , despi te the s ho r tcomi n gs men tioned above . An example of t hi s use is p resented later .
U S E O F R E S I D E N C E- T I M E D I ST R I B U T I O N S I N T H E
D ES I G N O F H OMO G E N EO U S R E A C T O R S
One advantage of u sin g t h e concept o f a residence- time distribution i n eval
uating tracer ex p e ri men t s has already b een discussed , namely , the p hysi cal insi ght in to the p roperties of t he system t hat is gained from the u s e of the escape probability II ( t ) . Just as important are the direct conceptual in sight s , w hich. can h e l p in un d er s tandin g reac tor desi gn problem s . T racer experiments are very useful i n organi zin g and evaluating alter native reactor de si gn s , a s well as in desi gni n g and evaluating pilot plant experi ment s . T hi s requires m ore than simple al go ri th mi c u s e s of the di s tribution in w hic h o n e tries t o comp ute performance directly from a resi dence time distribution . However , the conceptual framework w hich offers quide lines for scaleup and scale- down p roblem s is generally worth the effo rt . T he results in this chapter can be derive d ri gorously only for homogeneous re a c tor s , w hich are a s m all fraction of the cases met by the de si gn en gi neer . However , the gui deli nes for reac t or design derived from the m ( i n contrast to desi gn al gorithm s ) apply to het ero geneou s reactors , as lon g as one un der stan ds the basic principles . The differences are di scu ssed in a later section . Fi rst-O rder I sotherm a l Reac tions
Consider the case of a first- order i s ot her m al reaction system . set of co mpounds A j ( j
T here is a
1, 2, , n ) which can undergo composi tion c han ges such that the rate of formation of a compoun d A j is given by n
r(A.) ]
- L
-
i=1
=
( - k1]. . A ].
.
+
]1
k..A.) 1
•
.
( 23)
In other words , all reaction s between t wo compounds A an d B can be ex pressed by a set of reaction s
Shinnar
1 06
( 2 4) w here K A B i s t h e equilibrium con stant of t h e reaction A B . I f one s tarts wit h an inlet c o m po site C j o , th e ou tle t c o m p osition of a plu g - flo w reactor is given ( Wei an d P rate r , 1962) by the equation n-1
C . (t) J
=
a. .
JO
'E
+
r=l
a. .
Jr
( 25)
exp ( - l. t ) r
where Cj is the co n c e n t ratio n
of
componen t j
ex p r e s s e d as a mole fraction
tt o and tl r an d A. r a r e the (n - 1) nonzero eigenvalues of the s y s te m . j j are c o n s t a n t s that depend on the initial and e quili b ri u m composition of the system , an d t is the residence time in the ho mo gen eou s isothermal plug- flow r e acto r ( or the r eac ti on tim e in a homogeneous b at c h r ea c to r ) . E quation ( 23) an d i t s solution [ Eq . ( 2 5) ] are co m p let ely equivalent to a Consider a pro b le m in probability theory describin g stochastic s y st ems . lar ge set of particle s , w hich can ch a n ge their state in a discrete w ay . E ach particle is a s s oci a ted with a state Aj . The p ro b a bilit y of a sin gle p a r ti c le b ei n g in a s pe ci fic state A i i s Pj . I f the number of p a rtic l e s in the se t i s very lar ge , the p robability of a p a rt iqle to be in Aj i s exactly equiva lent to the mole fraction s of partic le s Aj in the sy stem , w hich in a first or d e r system i s p ro p o r tio n a l t o t h e co n c en tr at io n Cj . One can therefo r e look at Cj ei ther as a concent ration or as the p robability of a sin gl e particle ( o r molec ule ) t o be in s t a te j . T h e mathematical proof is gi ven in Shinnar et al . ( 1 97 3 ) . T h e reader unfamiliar with probability theory can i n terp ret E q . ( 2 5) as follow s . The concentration C jo of the p a rti c les entering the r ea cto r expresses , for a si n gle particle , the probabilit y of bein g in state j . T h e reactor changes this p rob ability into C j ( t ) . Each p a rti c le behaves here independently from its nei ghb o r and can u n dergo chan ges completely independent of an y other molecule in the reactor . This is t h e p hysical meani n g of a fi r st - or der reaction syste m . Once thi s is understood , some important results in re actio n engineering become clear wit hout further mathematical derivation . Consider a homo gen eo u s isothermal reac t o r w hich has a s t ea dy - st at e flow that can be charac te ri z ed by a residence - time di st rib u ti on f( t ) . T he inlet con c en tr ation is One can now p er fo r m t he follo win g t ho u ght experiment . An observer C jo stan ds at the outlet of t he reactor an d collects a ll the particles and sort s ·
t he m int o bi n s acc o r di n g to t h eir residence time . To d o s o , o n e has to divide t he residence time into intervals ll t . If ll t is small , all particles in t he bin , t to t + 6. t , e s s entiall y have the same re sidence time , t , in the As each molecule in t he bin behaves in dependently of i t s nei ghbors , sy s te m . the concentration in t he bin would be the same as at the outlet of a homo gen eous i sother mal plu g- flow re ac tor with residence time t . n- 1
J
p.(t)
=
C . (t) l
=
JO
c:t .
+
L
= r 1
a. .
Jr
( 26)
exp ( - A. t ) r
T he average concentration at the outlet of the reactor < c .> can b e obtained
by avera gi n g ov e r all th e
bins :
1
107
Residence - a n d C o n t ac t - Time Dis trib u tions
= l
j'0 f( t ) 0
[ a.
+
JO
�1
r= l
exp ( - A. t ) r
a. .
Jr
]
dt
( 27)
which is equal to
10
c.
n- 1
+
:E
a.
r=l
Jr
"'
f( A. ) r
( 28)
where f( A. r ) is t h e L apl ac e transform o f f ( t ) wi t h A. r sub stituted for the t rans for m variable ( s ) . E q u a tio n ( 2 8) is equivalent to the outlet of a hypothetical reactor described in Fi g . 2 0a , w hi c h con si sts of a set of paral lel tubular fl o w reactors with different residence times t . T he fr ac ti on of fluid e nt e rin g a tube with a r e si d e n c e time b e t w e en t an d t + A t i s f { t ) A t . In practice , a real reactor wo uld h ar dly look like Fi g . 2 0a , but t h e re is one case that is equivalen t . C onsi der t he case w here th e reactor feed consists of separate dro p l e t s encap sulated by im p e rm e abl e membranes which have a residence - time distrib ution F ( t) in the reactor {it i s equi valent to the re actor in Fi g . 2 0a ) . The in divi du al droplet s in suspension polymeri zation In a mixed reactor this is t r ue only in the li mi ti n g case behav e t hi s way . called se gr e ga t e d flow . In a firs t- order system , by definition , each mole cule can be considered s uch a sep arate dro p le t . T he well- known case of an ideally stirre d tank will illustrate E q . ( 2 7 ) . Consider the irreversible reaction A ->- B . In the case of a plug- flow re actor , C A ( t ) C A 0 e- k t and for a sti r re d - t an k reactor , a mass balance will giv e =
l
-
I �--------��--�, ------� ------------
------�1
L-
-
(a)
(b) Alternative flow models for Zwieterin g method of bou nding the conversion for a re action of nth or d e r : ( a) segregated flow ; ( b ) maximum mixedness . F I G URE 2 0
1 08
Shinnar
0
( 2 9)
or
For a stirred tank , f ( t ) i s here ( l l l ) e
E q . ( 27 ) ,
- tiT
and t herefore acc o rdin g to
( 30) In the mass balance [ Eq . ( 2 6) ] , i t was assumed that the concentration in the stirred tank i s uniform at any point in t he tank . For a linear system , a much w eaker assumption is s ufficien t namely that for residence - time dis Residenc4 tribution is exponen tial or that the escape probability is constant . time di s t ri bu ti o n s are linear p rope rty of the nonlinear model of the trans port p roc e s s in the reactor , and first- order reactions depend only on this linear p ropert y of the flow . In reality very few reaction s are truly linear , but it is a sufficiently good app roximation for m any system s . Many react io n s are also p seudolinear in the sam e sense that w hile the reac tio n rates contain nonlinear terms such as a Langmuir expres sion , they can be w ritten in a form ,
,
·
( 31) where Ki j is a mat ri x of linear reaction rate coefficients an d <j) ( Cjo ) is a non linear function of inlet con c en t ratio n an d pressure that can be taken out in front of the reaction rate m atrix . Although this is seldom completely cor For all those syste m s residence- time rec t it is often a good approximation . distributions are an imp o r t ant modelin g tool . ,
Dev i a tion of the R e s i dence- T i me D i s t r i b u ti on from K i netic E x periments
If one has a complex reac One can also use E q . ( 2 8) in t h e reverse way . tion system in which the ei genvalues A r chan ge over a si gnificantly wide ran ge , we can use it to recon struct f( s) . A s sume th at the values of A r hav e been measured from a batch e x p e ri men t or a plu g flow reactor , and one now has experimental values for all the concentrations C j at t he outlet o f an industrial reactor . One can then compute the v alue s of A r and use them to e s ti m a t e f( A r ) for the in dustrial reactor . One then has sev �ral values of t he Laplace tran sform f( s) from which a reconstruction of f( s ) can be attemp ted . I f the A r are wi dely spaced , the reconstruction i s quite goo d . T hi s approach will b e discussed in more detail later when we deal with contact - time distrib ution in hetero geneous reactors . -
...
109
Residence - a n d Con ta c t - Time Dis tri b u tions Opti m a l R eactor C oncepts fo r Fi r s t- O rde r R eact ion s
Iso thermal Reac to rs T h e conce p t s derived in the p recedin g s ection lead to some important con Sp ecifically , t hey clu sions with regard to choosi n g a reac tor confi gura ti on . provide gUi delines for cases in w h i c h a plug- flow reactor is th e preferred choice and for cases in w hich other reactor confi guration s will be p refera b l e . The p rincip l e that a first - order reaction system is c om pl e t ely equivalent to the case w here sin gle m ol ecu le s under go c han ge s from one state to othe r states , with defined probabilities , si m p li fie s many desi gn p ro b lems . Many op timal p ro b l e m s that are v ery difficult to deal wit h analy tically become in tuitively clear if one understan d s the fo regoin g p rin ci p le and applies it approp riat e l y . In the p recedi n g section it was s how n t hat in th e case of a fi r s t - order isothermal reaction system of arb i t rary complexity , the h y p o t he tical reactor in Fi g . 2 0a w ill give identical re sul t s t o a ny reac t o r havi n g the same residence- time dis tribution . T he outlet concentration is therefore t h e average o b t ain e d from several plug- flow reactors with di ffe r en t residence O nce o n e re co gni zes t hat all one needs t o d o is ave ra ge over a num time s . ber of p l u g - flo w re acto rs , o p ti mi zation is si m p le . C o nsider , for example , the case w here < C j ( t ) > for a specific v alue of t has a de sirable o p ti m um minimum or m axi m u m concentration of < C j> · Th e n assume that one has found a re ac to r wi t h a re si denc e - time di strib ution f ( t ) that gives an o p ti m u m
yield of Cj · In this reactor t he o u t l et concentration < c > is si m p l y the j average of all t h e different plu g - flow r ea ct ors . There i s a simp le al geb raic theorem that say s t hat in any s e t of n u mb er s with average X , there i s at least one member of the set e q u al or large r than X and a.t least one member equal o r smaller t han X . I f f( t) is a plug flo w , one has only a sin gle plug flow rea cto r wit h t = E f ( t ) = T an d which has a u ni q u e C j < Cj > . B ut if f( t ) is not a p l u g flow , o ne has in Fi g . 2 0a diffe ren t tubes with different outlet concentration . One o f these reactors has an o u tle t concentration equal or l a r ge r than t he average concentration < C j > and in one of these re actors C j is s m al l er ( o r l ar ge r ) than < c j > . T herefore , depending on whether one wishe s to maxi mi ze or mini mi ze < C j > , on e can alw ays fin d a tubular reac tor in Fi g . 2 0 wi t h a uni form resi den ce ti me w hich is as good or better than the reactor wi t h the a rbi t rary residence time f( t ) . T hi s is a si mp le proof that for first - order systems a p lu g - fl o w reactor is optimal i n the sense t h at o n e c a n d e si gn a plug- flow reactor wi t h a yi el d or selectively as high or hi gh er than any i sothermal reactor with ar bi t r ary residence- time distribution . It results in a very sim p le b ut powerful con c e p tual tool fo r reactor desi gn . I n all cases w here t he s y s te m is such t hat one can o b t ain the outlet co n c e ntrati on si mply b y av e r agin g over the bin s of differen t re sidence times , t hen , for any reaction system which has the p rop erty that there is an o p ti m um yield of one co mpoun d , a p l u g - flow reactor is t he best choice . I f , fo r other reasons a simple plug- flow reactor is no t fe asible or desirable , one w an t s a desi gn in w hi c h the resi dence - time di stribution ap proaches plug flow as cl o s ely as possible . For example , F ( t ) for a series of s tirr e d tanks approaches plug flow if the number of stirred tanks is reasonably lar ge . One can j u dge t h e re q ui r e d number of tanks si mpl y by =
110
Shinnar
the allowable deviation from plug flow .
S ta ge d systems in general h av e a
similar property and one can use the s a m e approach to estimate the number of stages re quired . For av e r a gi n g to apply , the reaction does not have to be s t ri ctl y first order . A ll one re qui re s is that t he reaction behavior of a mo le cu le is not affected by t he nature of i t s ne i g hb o r s , as di s cu s s ed in the p r e c e di n g sec tion . O therwis e , one needs to know not only how lon g a mo l ec u l e has been in t he system , but al s o who its nei ghbors were . If the outcome i s s t ron gly affected by the nature of its n ei ghbors , a residence- time di s t ri b ution can not p rovi de an answer for the o ut co m e . N or can one show that plug flow is op timum , although one can still get some g ui de li n e s w hich will be di scus s e in a la t e r s e c tion . In a si mi l a r way , if one looks at t he conversion of a first - order simple re ac ti o n A B wit h reaction ra t e
( 32) it is given by
a - a
eq
=
(a
0
- a
eq
)e
- ( k +k ) t l 2
( 3 2a)
A gain , the outlet concentration of a reactor wit h resi dence - time distrib ution f( t ) is obtained by av e r a gin g over the bins , - (kl k 2 ) t over f ( t ) . It is easy to see + w hich really me an s averagi n g e t hat a p l u g - fl o w reactor with the same average residence time t will h av e a low er a - ae an d hence hi gher co nv e r si o n t han a reactor in w hich the bin s have di f eren t t . ( Had the function been concav e , the opposite would For hi gh co n v e ri so n in a fir s t - o r d e r isothermal system , a hav e been true . ) plu g- flow reactor will h av e t he lowest res i d en c e time . A gain , o n e can evalu ate the permissible d e vi a tio n by some very si mp l e models ( such as a s e ri e s of s ti r r e d tanks ) an d jud ge the desi gn by measurin g the deviation from p l u g flow via a tracer ex p e ri m e n t . For small deviations the co effi ci en t of v a ri at ion y 2 of f( t ) is a good criterion . In the li tera t u re the P e clet number is oft e n used in stead and is app roximately equal to 2 f y 2 . M ore accurately , it is given by the expression ( see Shinnar a n d N aor , 1 9 6 7 ) .
T his is a convex function of t .
�
y
2
=
2[e
- Pe
- ( 1 - Pe) ] 2 Pe
( 33)
w hich for large Pe ap proaches
y
2
2 Pe
( Pe
+
00
)
( 3 3a)
B ut o n e should be careful not to use P eclet numbers smaller th a n 1 0 , be cause in m o s t c ases this is meanin gless , as t h e residence - time distribution will not fit a o n e - dimensional diffusion mo d el . I n fact that the re sults are not s e n s it i ve to the form of the deviation from plug flow is illustrated i n Fi g . 2 1 , w h e r e two cases are plot t e d . T he
111
Residenc e - and C o n tact- Time Dis t rib u tions 100
"I 9 0
. .q .... 0 z 0 Ul a:
� 80
'- I D E A L STIRRED
z 0 0
70
(a)
A ....'!.. a 1.0
2 .0
3.0
�
:I
X .q ::E
K
5.0
6.0
k
k
A _).. B � C
0.7
::t
4.0
-'---'---'--- ·_l __!.__L___L_--L._-.J
O. B
0 .q ..... CD
TA NK
0.6 0. 5 0.4
0.3 0. 2 0.1
0.0
(b)
IDEAL STIR R E D TA N K 1.0
2 .0
� kl k
3.0
4.0
5.0
( a ) Conv er si on o f a first- order irreversible reaction for differ residence - time distributions . ( b ) Selectivity of a con s ec u ti ve first order reaction for different residence- time distributuons . 1 . Mixed flow ; 2 , five C STRs in series ; 3 , one- dimen sional flow , Pe = 8 . 87 ; 4 , 1 0 C STRs in series ; 5, one- dimensional flow , Pe = 20; 6 , plug flow . FIGURE 2 1
ent
Shinnar
112
first illustrates an irreversible fi rst - order reaction at hi gh conversion , and For the second s how s the selectivity of a consecutive fir s t - order reaction . both case s , the limitin g cases of plu g flow an d a stirred tank are given and two models with equal Peclet number are given . O n e is a network of equal stirred tanks in series an d the other is a one- dimensional di ffusion system with a Peclet n umber chosen such t hat y 2 is equal for both cases . For y 2 = 0. 1 , the re sults are so close that it is impossible to plot the difference . For y 2 = 0 . 2 ( Peclet number 8 . 9 or 5 s tirred tanks in serie s ) the difference is already recogni z able b ut still s mall . One should not mi sinterp ret this statem ent . The results are alw ays sensitive only to the variance of f( t ) . In this case only models for which f( t ) has a reasonable similar form were compared . For such cases all models with eq ual y 2 will give si milar results , For computation it is therefore ad provided that y 2 is sufficiently small . vantageou s to use simpler models based on stirred tanks instead of diffusion , esp ecially since none of the models rep resent s the actual physics accurately . N o n iso thermal R eac tors
T he arguments in t he precedin g section can be easily extended to a noniso thermal reactor , p rovided that t he temperature profile is i mpose d from the outsi de and i s independent of the reactions . For example , consider an a rbitr a ry network of stirred tanks , each of which is homo geneous and has a uniform temp rature imposed from the outsi de , altho u gh eac h reactor may have a different temperature . The reactors are interconnected in an A gain , arbitrary manner . T here could be s everal inlets and outlets.. each enterin g molecule is equipped with a clock , but this time a more com plex clock i s c hosen that not only records t h e time spent i n a specific t ank , but also records each s tep in a specific tank in the order in which it occurs . ) If one looks at each clock , one finds a set of ti mes < t t . t 2 , t s , t l , t 2 , i dentifyin g each successive stay in tank i until it leaves the syste m . One again collects t hem into bins , but this time one insi s ts that not only the total residence time t is constant in each bin , but that all molecules in a bin have an i dentical clock history . T hus in each b i n not only are t he separate stays in each tank of equal len gth , but they occur in the same sequence . In each stirred tan k the reaction s are first order wit h fixed reaction rate , determin e d by t he temperature of the tank . This means that if one know s the state of a molecule ( or the concentration of each compound ) enterin g the tank and the len gth o f stay i n the tank , the p robability dis tribution at the time it leav e s the tank is known or can b e computed . T hi s means that if one knows t he whole history of a molecule in a bin , one kno w s the p robability of it bei n g in each one of the permissible states , w hich in reaction term s means that one kno w s the concentration of the dif For t hose w ho are somew hat uncom fortable ferent co mpounds in the bin . with p robability theory , thi s concept can be t ranslated a s follow s . Follow a blob of molecules that has an equal history . As t hey travel through each tank , the concentration of the b lo b changes in each tank in the same way it would in a plug- flow reactor wit h the same temperature and residence time . T here is now only a cascade of plug- flow reactors with different resi dence times and temperatures ( or reaction rates ) . Each bin represents a unique sectional plu g- flow reactor o f this type ( see Fi g . 2 2 ) . It was shown pre vio usly t hat each stirred tank can be presented in such a w ay that mole c ule s with residence time t can be lumped to get her and treated as i f they would be in a p lu g - flo w reactor of t he same residence time t . The avera ge outlet .
.
•
Residence- a n d
FIG U R E 2 2
113
C o n t ac t - T im e Dis tri b u tions
E qui v a l en t representation of a
particle
S ec t io n ally uniform plug- flow reactor . each section ; ti residence ti me of each s ec tio n
mal reactor .
,
history in a noni so the r
T i , t emperat ure of
.
is now the average of all those sectional plug- flow reactors clear that if one is all ow e d to average them , one can always fin d a single bin (or a plug- flow reaction with a s p e cial temperature and tim e sequence ) , which yi e l d s a result as good or better than t he network of stirred tanks . This p roof i s not limited to n e t wo r ks of stirred t an k s One can take any arbit ra ry flow and represent it by such a n etwork if one makes the stirred tanks small en ough This i s therefore t r u e for any arbitrary flow
concentration
and it is
.
.
system provided that : 1.
2.
T he re ac tion rates are locally first order in concentration . The temperature profile i s independent o f the reactions and i mpos ed from the ou tsi d e .
If co n di tion 2 is not fulfilled , as for example in an adiabatic reactor , one can no longer average . T he temperature of a tank will depend on its con centration , and therefore the r e ac ti o n rates affecting a sin gle molecule will depen d on the n at u r e of t he other molecules in the tank , which violate one Howeve r , as many r e a c ti ons are approximately first condition for a v er a gi n g order , ave r a gi n g p rovides a s i mp le guideline for reactor desi gn . E ven more important , it lets one understan d and reco gni ze the e x c ep ti o n s to the rule , in the cases w here plug- flow reactors are not optimal . T hese inglude , for example , all cases in w hich it is d e sir abl e to maximize the contact between produc t s and reactants . .
Bounding
M e t ho ds for N o nlinear
Reac tions
For a fi r st or d e r system , knowled ge of the r e si d e n ce tim e distribution co m pletely determine s t he pro d uc t co m p o s ition in a premixed i sothe r m al homo For nonlinear re ac tio ns this is not e nou gh as one n ee d s geneou s reactor . to know n ot only the history of a molecule , but al so the property and s tat e However , for a certain class of of the molecules in i t s close neighborhood . nonlinear r e ac ti o ns wehre the reaction rate is of the type -
-
,
rA
=
- kan
( 3 4)
Zwi et erin g ( 1 9 5 9) has show n t hat o n e can get ri gorous bounds on t he pos sible conversions directly from f ( t ) . C on side r again a s tirr e d tank w it h the simple irreversible re action A + B but t hi s time let the re ac tio n rate be second order. If the tank is ideally sti rr e d , a mas s bal anc e will give ,
- QC
Z
- VkC
2 A
=
0
( 3 5)
114
S h i n nar
and
( 3 6)
One can also namely , that
look
at the other extreme i de ali zed case
mentioned
stay together ( w hich Zwieteri n g term s se gre gated flow ) .
equivalent to the re actio n c on centratio n will be
in Fi g .
13.
T hi s case i s
I n each tub ular reactor the outlet
1
=
an d t h e
before ,
the enterin g flui d enters as separate encapsulated drop s that
( 37)
average
outlet concen tration will b e
>
S exp (
th e average
simply
over f( t ) ,
or
a
Ei i s t he exponential inte gral . ferent results . However , if k C A O t small ( see Fi g . 2 3 ) .
00
1
=
kC
w here
t
AO
not
T h e reason for the difference is that
1
=f
-
very large ,
for a
e
0
that the two
N ote is
E.
-u
u
0 /u
lead t o difference
cases the
( 38)
dif is
s e c o n d o r der reaction the
p robability of a molecule of substance A r eac tin g depends on its p robability
molecule of A . In the segregated case , t his is larger than in t he mixed case , a s in the latter t h e enterin g molecule is immediately exposed to p roducts t hat reduce C A in its surround in gs . K nowle d ge of the resi denc e - time di strib ution alone is here insufficient to comp ute C A , as one needs to know w hat happened not only to the mole cule but to its neighbors . H o we ve r , for this simple cas e of secon d - order reaction , Zwietering was able to show that ideal mixing a n d s e gregated flow of
bein g close to another
probability
condition s are extremes and that all other possible mixin g sit uations for
f ( t ) is the
w hic h
C
A
For n
r
A
(ideal
=
=
s am e will res ult
mixi n g )
> C
A
in an
( s e gregated
outl et concentration such that
( 3 9)
flow )
1 / 2 or - kC
A
1/2
( 40 )
the limits become
2
( � ) (� � kt
2
C
AO
2
kt
2
2
C
AO
( 41)
115
Residence- and Con tac t - T ime Dis t ri b u tions
0.8
0.4 0.2 0.0
0
2
4
3
5
-
-
F I G U R E 23 C o nversio n of a sec on d o r der irreversible reaction ( A B) in a sti rre d tank as a function of time for segregated an d completely mixed flo w .
an d
C
[ (� )
l ( s e gre ga ted flow ) = C Ao A
+
kt
2
2
C
2
-
( 42)
AO
Note that C A for t he i deal stirred tank is l e s s than C A for se gregated flow . Here the presence of molecules of A reduces the r eact ion r a t e a n d mi x in g However , one has to be v e ry careful wit h products improves conver sion . with such arguments w hen applie d to reaction s of fr ac tional order . T h ey arise nor m ally from Langmuir- ty p e re ac tion r ate expres sions . Expressin g them as a fractional order is correct only over a lim it ed ran ge of con c entra tion s . T here is a high probability t hat in in t e gr ati n g the results for Eq . ( 40) , one m ay go o u tside the ran ge w here this is applicable , an d t herefore T he one must verify that the final result is not se n siti ve to this mistake . results for a n on line a r reaction t herefore dep end not only on f ( t ) but on a mo re complete descrip tion of the mixin g phenomena which Zwieterin g t e rms
micromixin g .
Swieterin g has also s hown that these results can b e extended t o any arbitrary re si dence - time di stribution as lon g as the r e ac ti o n is a si mple re In that case C A I C A o will al way s be bounded by two ex tre me cases : the case of segre ga ted flow given in Fi g . 2 0a and another Where case defined as maxi mum mixedne s s , w hic h i s illu strated in Fi g . 20b . as in Fi g . 2 0a ea c h tubular re ac t or is s epa r a t e in Fi g . 2 0b , t here i s inter mixing betw een them and the feed inl e t s are arranged such that t he expected action of order n .
,
S h in nar
116
future life is constant for all molecules in a given cross section . P h ysi c al ly , one could reali ze t he a r r an gem en t of F i g . 2 0b b y d e si gnin g a pl u g flow r e actor in w hich additional feed is i ntr o d u ce d at different poi nt s along the t ub e . For a r ea c tio n of order n the two limi t s or bounds on conversion are se g r e ga t e d flow and maximum mixedness . For the case of segre gated flow , 00
J
[1
+
( n - 1)RJ
111
-
n
f ( t ) dt
( 43)
0
- kt R = -�1-n
C
AO
For the case of m aximum mix ed n e ss , < C A > / C A O h a s to be o b t ai n e d by solv
i n g the differential equation
( 4 4) w here A is an i nt e gr a ti o n variable and A ( A ) is the escape p robability a s de fined by E q . ( 8 ) . E q ua ti on ( 44) can be int e grated numerically , although for certai n valu e s of n and certain values of 1\ ( :>.. ) it has analytical solutions . Here one sees another useful p roperty of the escape p robability and why one w a nt s to m e a s ur e the re sidence- time distribution in such a w ay that A ( t) can be comp uted reliably . For reactors close to plug- flow reactors , the bounds are very close and one does not have to worry about micro mixi n g w hi ch i s im p ortant mainly in stirred- tank reactors . S t ri c tl y speakin g , the results of E q . ( 43) apply only to si mple reaction of o r der n , in homogeneous ideally stirred reactors , w hich is a rather rare desi gn p roblem . However , the concepts of boun di n g t he outlet co m position of a reactor is an e ssential tool of reaction engineerin g . I t i s really a fo rm of p e rfor min g a s e n siti vi t y analysis on a de si gn . Consider the example o f a A priori one knowns that this is an i deali zed model t ha t can sti rre d tank . H o wever , o n e c a n approach i t in t he sense that the not b e fully reali z e d . time a droplet of feed spends in the reactor before it becomes t ot all y mixed i s very s mall com p ared to the tot al residence time ( Ev an geli s t a et al . , 1 9 69b ; S hinnar , 1 9 6 1 ) . O ne also kno w s t hat durin g scaleup t he residence time s t a y s constant , w hereas the mi xi n g ti me always i nc re a se s ( Evan gelista et al . , 1 96 9a , b ; Shinnar , 1 9 6 1 ; W ei n s t ei n and Dudkovic , 1 97 5) . For a se co n d - o r d e r sin gle reaction , Z wieterin g' s results show that one p robab ly does not need to worry about this , a s the completely mixed case is t he worst ( least de T here are many ot her cases for which this i s not true . One si rable ) cas e . wants to know how important s uch deviation s from complete mixin g are . To answer t hi s , the two limitin g cases for any arbitrary reac ti o n system are d e ter m i n e d . If the reactions are complex an d nonlinear , t he se two limiting cases ( s e gr e g at ed flow and complete mixing ) will not give rigorous bounds b u t will indicate i f t he system is sensitive to mixin g and w hat p enalties are i n c ur re d for in com p l e t e mixin g . In scaleup , one cannot achieve se gregated flow , so the case of complete mixi n g must be s ati s fac tor y or the desi gn must be chan ged . If the s y s t e m i s sen sitive t o mixin g , one can estimate t he per missible mixin g tim e by u si n g models for incomplete mixin g that si m ul at e
Res idence- and C o n t ac t - Time Distrib u tio n s
117
small deviation s from in co m p le t e mi xin g ( E v an geli s t a et al . , 1 9 6 9a , b ; B elevi
et al . ,
1981 ) .
I f , in a sti rr ed - t an k reactor , s e gr e ga t e d flow leads to improved re sult s ,
a plug- flow reactor will n orm ally be even better . If one r e qui r e s the stir rin g , one can a ppro ach plug flow by putting several such s ti rr ed tanks in se ries . E xperiments that de m ons t r a t e the s e n si tivit y to d e vi ation s in mixin g can al so be performe d in a pilot p l an t . For ex amp le , one can desi gn a
pilot plan t similar to the one shown in Fi g . 2 4
des pi t e the fact that the T here hav e been a number o f models propo se d that look at m icromixin g for more comp lex flow co n di t ion s over a w i d e r ran ge of mic ro mixi n g [ see , e . g . , Weinstein and Adler ( 1 9 67 ) ; N g a n d Rippin ( 1 96 5 ) ] , T he s e models , which are reviewed in detail in N auman ( 1 98 1 ) , are more in t he nature of l e arnin g model s than de si gn models , an d are th e r e fo re outsi de t he scope of this ch a p t er . Si milar c o nsi d er atio n s app ly to p ack e d - bed and fluid- bed reactors . For nonlinear r eac ti o n s the only safe w ay to sc al e them up is to approac h plug flow as clo s ely a s p o s si b l e . A gain , one wants to know how closely plug flow m u s t be appro ac hed and w hat the p e n al t y is for s mall deviations . One can again p e r form e x p e rim e nt s w hic h ex p e rim en t all y check the sensitivity of the reaction system to d e v i ati ons from plu g flow an d desi gn the reactor accordingly . H e re re si dence- time di s t ri b u tio n s are directly u seful , since for small de v i atio n s from plu g flow , one can estimate the deviation fro m A si mple model will p l u g flow fro m the tr ace r response by m ea s uri n g y 2 . gi ve reasonably reliable bounds as to the possible impact of such small de viation s , even i f t he reactions are hi ghly nonlinear . If the sen sitivi t y i s large , o n e s hould take a safety coefficient with re s p e c t to t he allowable co efficient of v ari a t ion . T hi s is i l lu st r a t e d in Fi g . 2 5 , w here the effect of small dev ia tio ns fr o m p l u g flow on s e v eral reactions are comp ared . One i s simply the irreversible r e act io n A + B of se co nd ord e r , and t h e other is a nonlinear ca s e very sen sitive to mixin g . In both cases th e re sult s a re in distin gui s hable , and for y 2 = 0 . 2 ( five stirred tanks in series , or a Pecl e t number of 1 0 ) the results are r ea so nab ly close . In both cases we also giv e the limiti n g cases of p l u g flow an d a stirred tank to indicate the sensitivity to back mixi n g . T he comments made with re ga r d to Fi g . 2 2 apply here final reactor i s c omp l etel y mixe d .
FIGURE 24
Si m ula ti o n of finite mi xi n g in a p i lo t p la n t .
118
Shinnar
equally w ell .
T he results are not unique functions of y 2 .
How ever , for
s m all deviations from plug flow , t he results are not model sensitive as lon g
as the residence - time distri butions are rea sonably similar . For simulation p urposes , a model based on stirred tanks is therefore preferable , as it is T he fact that y 2 alone i s not s ufficient t o charac much easier to compute . teri ze the system unless y 2 i s very s m all is illustrate d in Fi g . 2 5c w here two ot her models with y 2 0 . 1 are compared to a serie s of stirred tanks . For a reaction sen sitive to mixin g , such as t he one show n in Fi g . 2 5b , the differences are very lar ge and a m uch smaller y 2 is necessary for them to become ne gli gible . Such reaction s t he refo r e require extra care in scaleup and a close app roach to plug flow ( or a stirred t an k ) . =
80
70
� 50 u
40
30
3.0
2.0
1.0
4.0
(a) F I G U RE 2 5 ( a ) C onversion of a second - order irreversible reaction for dif ferent flow models . ( b ) C o m p a ri so n of p roduct yield in a consecutiv e non Reaction A + B + C fir s t order ; linear reaction for different flow models .
reaction A + C + D second orde r . Inlet concentration of A , 1 . 0 ( di m ensio n less unit s ) . 1 , Mixed flow ; 2 , five C S T R s i n series ; 3 , one- dime nsional flow , Pe 8. 8 7 ; 4 , 1 0 C S TRs in series ; 5 , one- dimensional flow , Pe = 2 0 ; ( c ) Same as in p art ( b ) for different flow models : 6 , p l u g flow . =
(I )
VPF
+ =
[=:J
+
Ll..J
0. 69 V , V ST
=
+
(II )
0. 31.
1 0 stirred tanks in serie s ( I I I )
....
l.LJ
....
c:=J+
119
Residence - a n d C o ntac t- Time Dis t ribu tions 0. 0 5
0
A .!. e.!.c A + C 0"1
0.04
w w u.
z c(
D
0.0 3
w ..J
0 ::iE 0
......
0.02
0 u.
0 ::iE
0.01
/
;y
/:/_ .
0. 0 0
...... -
- -��
-- - �
-·
1 .0
2.0
3.0
4.0
L-----'-----'----.I..... ---'---___.___.J
(b) 0.04
-
/ �Pl.� ./
II) w ..J
--
- '"4
TIME -
----- -------�
A 2. e ....!. c A +
c ..QJ.. o
0.0:3
0 IL 0 Ul w ..J 0
::!: 0. 0 2
0.01
0.00
(c)
1 .0
2.0
F I G U RE 2 5 ( Continued)
3.0
T I ME -
4.0
120
S h in nar
T here is another area in which the concep t of a residence - time distribu tion is useful in o r ganizi n g ones thinkin g and experimentation . T h inking in terms of residence times and joint p robabilitie s is helpful i n de ci di n g what reactor types t o choose . For all linear and p s e u do lin e a r system s , iso thermal as well as nonisothermal , plu g- flow reactors are alw ay s p re fer able . The case w here bac kmi xin g or a stirred tank has a po si ti ve influence is Ex w hen there are complex interaction s betw een p roducts and reactants . amples a r e autocatalytic reaction s , autothermic reac tions ( hea t c a n be con sidered a prod uct ) , or reactions of the types A + B
+
C
A + C + D If D is d e si r abl e , a mixed reactor is p r e fe r abl e , as can b e seen from Fi g .
2 5 . I f D is undesirable , a plu g- flow r eacto r is p re ferable . of thi s reaction di scussed in R enick et al . ( 1 9 8 2 ) : Methanol
+
olefines
+
aromatics
A r om a tic s + m e t hanol + durene
+
An example
p araffin s
I f D ur en e form ation i s to be avoi de d , the reactor should b e as close as possi bl e to plug flow . T hi s re s ult can be derived wit hout reference to residenc e - time distribu tion , but thinki n g in term s of resi dence time p ro vides a useful fram e w ork . T here are a num b er of ca s e s in w hich the desirable confi gur ati on i s neither p l u g flo w nor a sti rr e d tank . Combustion is an example . B ut here one can often b reak dow n the optimal or desi r e d flow sys t e m into confi gura tions composed of w el l - d efi ne d flow s . I f , o n the other hand , the flow i s complex , residence- time distributions may pr ovi de dia gnostic test s , b ut one can neither model nor scale such cases sol e ly on t h e basis of residence- time distributions . It s hould be emphasi z e d that at the p resent state of knowled ge , t here is no w ay that one can reliably scale up a reactor if the reactor is sen s itiv e to the form of f ( t ) and the r e ac tor does not approach one of the well defined i deal cases . ·
H E T E R O G E N EO U S R E A C T O R S A N D
M U LT l P H A S E S Y S T E M S
Prope rties o f T racer E x peri m e n t s i n M u l ti p hase S y s tem s
Residenc e - time distrib utions p r ovi d e a powe rful method for the a n al ysi s of ho mo g e ne o us i sothermal reactors . Unfortunately , very few real reactors used are t r uly ho mo ge n eous . Those few that are seldom p rovide se riou s difficultie s in desi gn or scaleup . The majority of t he re act ors used are heterogeneous , eit her gas - solid , gas - liqui d , or triple phase . Catalytic re actors are almost alway s heterogeneous , even in homo geneous catalysis , w he re most of ind ustriral reactors deal with gas - liquid reactions . If t h e use of re si den c e - tim e distrib utions was restricted to homo ge n eo u s Luckily , m o s t of t he reactors , t heir usefulness woul d b e rather limited . concepts derived in the pr evio us sections can with some care , be applied to hetero geneous reactors . One must , ho wever , be careful to take i nt o account the hetero geneous nature of the system . T his rules out many of
1 21
Residence- a n d C o n tact- Time D is t ri b u tions
the algorithmic uses of residence - tim e di st rib utions . One cannot simply m easure a residence- time di stribution and predict the performance of a com Even t he concept of a residence- time di s t rib u ti on is not sim plex reactor . ple any more , as different reactants m a y have totally different residenc e time di stribution s . I n a hetero geneous reaction one cannot measure reaction rates and re sidence - time distributions i ndependent ly and reliably predict re actor performance even for first - order system s . B ut the main uses focu sed on in the precedin g section remain .
1.
T he intensity function is still a useful tool to dia gno s e maldistribu tions in t he flow and provide clues to the nature of the flows in Of special importance is t h at it allows one to check how t he system . closely a desi gn approaches plug flow (or an ideally stirred tank ) It is also usefu l in cold- flow models , w here it c an be used to test the efficiency of flow distrib utions or baffle s . T he examples given earlier were fro m hetero geneous reactor s Residence- time distributions still provide a powerful conceptual tool for desi gn and scaleup . A lthou gh often one cannot get accurate p rediction s , one can derive a better un d erstandin g of the scaleup p roblem . Resi denc e - time distrib utions i n multiphase systems allow m easurement of t he hol dup o f each phase . .
.
2.
3.
T here are some imp o rtant differences between tracer exp eri m en t s in homo geneous and he t ero ge n eou s sy stem s . It is i mportant to understan d these differences not only in evaluating tracer experiments , but also because they are related to desi gn and scaleup . In a hetero geneous system , di fferent compounds m ay have stron gly different resi dence - time distribution s . In a homo geneous reactor t he difference in residence- time distrib utions between different compounds i s , i n most cases , ne gli gible . T his is especially so if T he only exceptions are slow laminar flow s , the mixing process is turbulent . w here mo lecu l ar diffusion will be important . T hu s in most homo geneous re actors the nature of t he t racer compounds is of small importance . On the other hand , in heterogeneous system s , there can b e lar ge differences be tween the response of different tracers , and therefore between the respon s e o f a tracer and that of a reactant . Even different compounds in t he feed or different products may have different residence- time distribution s . An example o f this is shown in Fi g . 2 6 ( see also O rth and S c h ii ge rl , 1 9 7 2 ) . Fi gure 2 6 i l l u st rates data taken from a pack e d - b e d Fi scher- T ropsch reactor in which CO an d H 2 were reac te d over a Ni c ataly s t . T he reactor w as operated at s t eady state and a step in p u t of CO containin g 1 3 c w as intro T he helium concentration w as duced to gether wi th a step input of helium . T he re sponse o f kept very small so as not to di sturb the steady state . the 1 3 c in t he C O is plug flow with a lo n ger residence time , which show s that the CO is adsorbed but t hat the interchan ge of the adsorbed CO with the CO in t he gas p hase is very fast . On the other han d , both the C H 4 and C 3 H 6 p roduct b ehave like a stirred tank i n series with a p lu g flow . This indicated not only stron g adsorp tion but that , in this case , the inter change b etw een the adsorbed phase and the gas p hase for t he pro d uc t i s T he re slow and has a characteristic time similar to t he residence time . sult s could also have been the reverse ( slow i n t erchan ge of the reactant and fast interchange of the p roduct ) . In a reactin g system the concept of a residence- time distribution itself b ecomes rather c om ple x to define .
1 22
Shinnar
,/H e
1.0
I I , I I r I r r I
0.5
0
r I I I I I
0
/
100
200
30 0
400 I ,
sec
FI G U RE 2 6 Response of a packed- bed Fi scher-Tropsch reactor , fed with an d H 2 , to a step input of 1 3 C O and H e . Reactor kept at steady state . ( From Biloen et al . , 1 98 3 . ) CO
One can always define it with respect to specific elements such as C in Fi g . 2 6 ( o r molecular combination s that d o not chan ge such a s benzene rings ) , but the only way to measure this would be by means of proper isotope markers . In Fi g . 2 6 the tracer response is given separately for 1 3 C O and 13c H 4 . From the ori ginal data one could have measured a tracer response for 1 3c , which would give a residenc e - time distribution for carbon atoms . I t i s a proper residence - time di stribution , containin g valuable information , but its interpretation is more difficult . It is not only a function of the flow pattern but also a function of conversion ( or catalyst reactivity ). To measure maldistribution in flow , a nonreactive tracer is often p referable . T he average residence time of a sin gle tracer is not necessarily equal to total reactor volume divided by the flow rate . I t is therefore hard to estimate independently the average residence time , which due to adsorption or similar phenomena , can vary by a factor of 5 or more betw een different tracers . Furthermore , not all tracers give linear responses . This makes the choice of a proper tracer much more critical than in a homo geneous system . Residence - time distributions are by definition linear func tion s . At steady state all compounds in a reactor have a measurable resi dence-time distribution re gardless of the nonlinearity of t he flow , nonlinear adsorp tion phenomena , and so on . If one can properly mark a compound ( e . g . , usin g an i sotope ) , the tracer experiment will result in a residence time distribution . B ut if one introduces a tracer that is not present in the system , it could behave in a nonlinear way , and the result could be very dependent on the amount of tracer introduced . Furthermore , one does not always want a tracer that behaves similar to the reactant . For example , in some cases w here one is studying reactor internals in a packed bed , non adsorbing tracers are preferable , as they are more sensitive to a maldistribu tion in the flow . This can be seen from Fi g . 2 6 . A heli um tracer would have given a much better indication of a maldistribution in gas flow than using 1 3 c o and measuring the total amount of 1 3 c leavin g . The proper choice of t racers is discussed in more detail in the next section . Residence- time distributions cannot be used directly to predict outlet composition even for linear reaction systems . Reaction rates in catalytic reactors are measured and reported on the basis of either reactor volume ,
1 23
Residence- a n d C o n tact- Time Distrib u tions catalyst volume , or catalyst w ei gh t . same time scale as t he re si de n c e ti me tem t hi s re la ti o n s hip i s not v ali d ( due solubilitie s , etc . ) , and the di ffe re n c e
In a h o mo ge n e o u s system this has the
distribution . In a hetero geneous sys to adsorption , ab s orp tion , different in time sc ale can be a fac tor of 5 or more . T he r at i o is di ffe r e n t for different reactants . Consi der , for exam ple , a p a cke d b e d reactor . T he estimat e s of ki ne tics are based in most cases on overall re si d e n c e time s of m ole reactant}lb catalyst p e r hour . T he real residence - time di st ri b u ti o n of the gas i s in most cases unknown b e c aus e t h e ho l d up of t he gas depends on adsorption . If one could really tag a re actant one would not nec e s s a ril y get p l u g flow in a tubular laboratory re actor ( see Fi g . 2 6 ) . However , t h e ki n e ti c d a t a u n d e r l yi n g t h e s cal e u p assumed plug flow in t h e lab reactor . T he rate ex p re ss ion s are really not molecular kinetic e x p r e s s i o n s d e s c ri bi n g real molecular events but r ep r es ent overall re l a ti o n s allow i n g one to scale up a lab reactor to an industrial packed b e d . T his ap proa c h works very well even if the reactant is stron gly adsorb ing and t he reactions are h i gh ly nonlinear . The fact that for a hi gh l y ad so rbi n g reactant t he real residence time of a molecule is , in reality , far from plug flow , a n d different for all react ant s , does not affect the u seful ness of s u c h overall kinetic relations for steady - st ate desi gn . I t m ay how ever , s t ro n gl y affect t h e d y n a m i c behavior of the reactor and one has to be ca r e fu l w hen u sin g r at e e x p re ssio n s o b t ai n e d fro m s t ea d y s tat e e x p eri men ts to de scribe the dy n am ic b ehavior o f a reactor ( S hinnar , 1 97 8 ; B i loe n et al . , 1 98 3) . T his al so p revents one from u si n g such methods as t he boundary methods of Z w i e t e ri n g unless once a gain o n e i s d eali n g with s m all p e rt u r ba tions . In c a t al y ti c reactors , reactivity can c ha n ge wi t h p o si tio n even i n i so t her m al rea c to r s I n so me c at al y t ic re ac ti on s e s p e ci ally at hi gh t em p e r a tures , reaction s occur in the g a s p h as e as well as at the ca t al yti c s urface . T here is t h e re fo re no unique co r re latio n between residence time and t he state of t he molec ul e , w hich i s a p roblem already en co u n t ere d in t he case of the nonisot herm al reactor . While a residence- tim e di strib ution c an gi ve i m port an t information about t he flow , one w o u l d need ad ditional information to p r e di c t t h e o u t l et co m pos i t i on , even if the reactor is i so t h er m al and first ord er T he co n ce p t of a re si d e n c e t i me distribution is s till hel p fu l in the sense outlined in t h e b e gi n ni n g , as lon g as on e is careful to reco gni ze the di ffe r ences . This wi l l be di s cu sse d in detail at t h e end of t he s ection . Firs t , some techniques t hat a re e s p e ci ally applicable to h et e ro ge n eo u s reactors will be discusse d , n a m e ly m u lti ple tracer e x p eri m e n t s and contact- time distributions . -
-
,
-
,
.
,
.
-
,
M u l t i p l e T racer E x pe r i ments A s in hetero geneous s y s t em s , different compounds may have di ffe r en t re
sidenc e times . I n de p e n d ent knowled ge of the re si d e n ce - ti m e di stribution of sev er al tracer compounds gives a b e tt e r description of the system . I f the feed cont ains different compounds , one may wish to use m ar k e rs for e ac h of t he m to study t h e system . One can also learn a consi derable amount about the s y st e m by u si n g a n u m b e r of different t r ac ers even if t h e y are completely di ffe re n t from the re ac t a nt flui d ( Shinnar et al . , 1 97 2 ; N auman , H o w e v e r , one is really measurin g the 1 9 8 1 ; Ort h and S c hli g er l 1972) . residence- time di s t ri b u tio n of the s p e ci fic tracer , which is not neces sarily ,
,
Shinnar
1 24
representative of the system .
If several t racers of stron gly varyin g
p rop er ties are utili zed , the residence- time distributions
of t he comp ounds in the system can be bracket e d . One should note , how e ver , t hat even i f one uses a n i sotope of a reactant , o n e still cannot call thi s a resi d ence - tim e distrib ution of t he system , as in a hete ro geneou s system different reactants may have totally di fferent residence - time distributions . One does not alw ay s want a trac er that behaves like the reactant flui d . Consider , for e xamp le , a p acke d - bed reactor . To measure maldistributions of the gas flow , one w ant s a tracer that does not di ffu se into the cataly st and d o e s not get adsorbed . I n this case the t ransport pr o cesses i n si de the cat al ys t parti cl e s an d a d so rp t ion p rocesses are irrelevant fo r scaleup or reactor performance in the s e n se that one deals with s c ale u p of a p i lot plant The main p ro b lem o n e n eeds to co n s i der with i dentical cat alyst p a rticl e s . is t hat mass t ran s p o r t processes to and from the particle to no t chan ge as the linear velocity in the large reactor increases . T racer experiments are not sensitive enough to check this p rop e r t y , but t here are other m etho ds to e n s u re t hi s . The main use o f a tracer experi men t is to test that t here are no maldistributions in the overall flow . An exam pl e of t hi s was shown i n Fi g . 1 6 . I n a packed bed , i f the re are maldistributions , one can use the result of a t racer exp eriment with a no n ad sorbing tracer to e s ti m at e their impact on c o n v e rsion and p ro d uct distribution . In t rickle beds one must worry about both liqui d and gas distribution , and would there for e w ant two t racers , one for each p hase . In all these c a ses , A ( t ) is the most sensitive way to c heck for a mal dist ri b u tion , but in a trickle b�d one m u s t Stron g d evi a ti o n s from pl u g flow can occur due to fluid dy n ami c be careful . effects an d o n e t here fo re needs a goo d reference case ( Mi l ls and Dudukovic , 1981) . In fl ui d - bed reactors the situation is more complex . If th e solid p hase is not s t atio na ry b ut is fed an d removed from the bed ( as in the treatment of solids in an F C C cracke r ) , one may w ant to use a solid tracer to check if the solid has the desired residence- time di st ri b utio n . In a sin gle stage one pro b ab l y want s complete mixin g of the soli d s or a constant A ( t ) , w h e rea s in a m ul tis t a ge bed and for t reatment of solids , one w ant s the residenc e - time di st rib ution for the s oli d to corr es pon d to a s e ries of stirred tanks , or to ap p ro ach plug flow . I n some t ric kl e beds , part of th e feed ev ap or ates . In t his case , to study the flow p r o p erl y , it is recommended that b ot h an evaporatin g and nonevapo rating fracti o n of t he liq uid be tagged . An e x am p le of such a multiple t racer experiment is shown in Fi g . 27 , which illustrates r e s ul t s ob tained in a hydro t re at e r for fuel oi l ( Snow , 1 9 8 3 ) . T h r e e tracers a re used . T h e C 1 6 compound rep resents The C g co mp o un d enters in the gas phase . a comp ound that under t he conditions of t his h y dr ot reat er i s p resent in both T racer phases , whereas the C 3 2 c o m po u n d stays m ai n l y in the li qu i d phase . Re results are gi ve n for two react o r s op erat i n g under similar conditions . actor A co ntai n s a 1 / 8 - in . e x t r u d at e hyd ro t r e atin g cataly s t , w hereas react o r B op e rat e s with a 1 / 1 6- in . e x t r uda t e . Reactor B s how ed a better perform ance at hi gh conversion , allo wi n g a 5 0 % hi ghe r through p u t at 99 % c o nver sio n . T he results gi v e a good i ll us t r a tion of t he advanta ge of u sin g proper multiple tracers . T he most i nte re s ti n g information in Fi g . 2 7 is not the for m of the individual t racer respon se but the relative p o si tio n of the i n divi d u al tracer responses . In a t rickle bed , o n e would expect that t he residence tim e of t he l qi ui d is larger t han t hat o f t he gas . The results bear t hi s out , as the average residence time of th e C 32 response is larger in b o t h reactors then in th e C 8 tracer . In r e act or B the C 32 tracer appears 3 min a fte r the C g t race r . However , looking at Fig . 2 7a one notices imme d iat ely
125
Residence- a n d C o n tact - T ime Dis t rib u tions . 1 6 .------
-- - -- · -
Unit A
f(!)
.1 2
I I I •
"'
lj /�
.0 8
I '
�
.04
I•
0 0
4
8
12
20
16
. 3 2 .-------,
f(t)
Uni t B
.24 .1 6
.08
0 0
(a)
4
8
12
16
20
F I G URE 27 Multiple tracer experime n t s i n t w o industrial fuel oil hydro treaters . ( Private communication with A . I . S now , Arco Petroleum C o . , 1 9 8 3 . ) T hree compounds tagged w i t h 1 3 c introduced simultaneously in feed
(pulse exp eriment ) and measured at reactor outlet . ( a ) R esidenc e - time density function ; ( b ) inten sity function ( escape probability ) of unit A ; ( c )
intensity function ( escape probability ) o f unit B ; ( d ) cumulative residence time distribution ( unit A ) ; ( e ) cumulative residen ce - time distribution ( unit
B).
1 26
Shinnar
0. 3 0
U nit A
ACtl
0 .1 5
0.00 6
0
12
18
(b) ACt>
Unit B
0.60
0. 3 0
./· - -- ------�.!2_
tl
�--
_ __
.l.._ _J._____JI... .l.._ ...- --1---...J o.o o L-�i_____.J.___!__ _L_...._....
(c )
0
FIG URE 27
6
( Continued )
12
18
Res idence- and C o n t ac t - T ime Dis t ri b u tions
10°
1- F
127
r---��----� Unit A
(d )
10 3 �--L_ 0
6
(e) FI G U R E 2 7
12
18
L___L___L___L___ L---L---L---L-�
__
( Continued )
128
S h i n nar
that the c 32 t rac e r in reactor A ap p ea r s about 1 mi n before the C s t rac e r . This s how s a si gni fican t difference between the two reactors an d s how s th at so m e t hi n g is w ro n g with reactor A w hich explains its lower p e r fo rm a n c e . If one ex ami n es in detail at t he e sc ap e p r ob a bility ( Fi g . 27c and d) , one also notes that the C 3 2 re s po ns e shows a s tron g maldistribution in reactor A and good liquid di stribution in re ac to r B . I nterestin gly , react or B has a m al di s t rib utio n in the gas flow ( C s ) which is n o t evi dent in reactor A . B u t it is ap p ar en t ly not si gni fi c an t eno u g h to cause p rob l e m s . T hese tracer results contain some very i n t eres t in g information about what h ap pe ns i nside the se re ac t o r s , an d h avi n g sim ultaneous information from both a liquid and gas t racer gre atly adds to t heir value . However , the i n t e rp re t ati on of these resul t s is in no w ay uni q ue , as the mal di stribu tion of t he C 3 2 re s p o n s e in reactor A co ul d be due to s e v e r al causes . It co ul d be c a u s e d b y a mechanical p robl e m re s u l ti n g in a nonuniform liquid di s t ri b u tio n , or b y inherent flow problems , or by the di ffere n t catalyst si ze . Some reactor m od eli n g could indicate w hi c h of these e x p l an at ion s i s mo re likely . It is i m p o rt an t to note that in such a m ul ti phase system , there is n o w ay to deduce from a single t racer experiment if a. maldistrib ution s uch as the C 3 2 response in reactor A ( Fi g . 2 7 ) is caused by n bad di s t ri b u tor or by b y pa s s p henomena inherent in the flow regime of t hi s s p e ci fi c r ea c t o r or by ph e no m en a inside the catalyst particle . T rickle beds , by their n at u re , do not sc al e equally well in different flow r e gim e s , which are d ete rmi n e d by various flow p ar a me t e r s , such as gas v e lo ci t y and density , liquid p roper t i e s , a nd liqui d - to - ga s ratio . This problem w a s alr e a d y e n co un t e re d in the disc u s sion of the fluid bed , and it is i m p o r t an t to reali ze this . However , often on e has two reactors o p e rat i n g under similar con di tions and , in s uc h a case , a difference in tracer re s po n s e cle ar l y indicates a mechanical problem . I n s t u dyin g t he gas flow in a fluid bed , one faces a p roblem not en co untered in pa c k e d beds . If the r e a c t an t is a d s o r be d , it will move wit h t he s oli d . I n an i so t he r m al or adiabatic packed bed , i f a pilo t plant and a large p l an t have the same residence- time di st ri b ut ion for a non ad so r bi n g t racer , one would expect them to behave si mi la rl y . I n a fl uid bed one can no t make t h e same as s u m p tio n , a s the mixi n g p rocesses o f t h e solid and the a d so r b e d reactant with it are di ffe re n t for t he t w o c as e s . T h e only ti m e that one can r eli a b l y scale a fluid b e d an d p r edi c t i ts p er for m an c e from a pi lot p l ant i s w here in bo t h cases the g a s flow approaches p l u g flow as closely as pos sible . One way to a c hi ev e this is by u si n g baf fle s in t he fluid bed . B affles here have a double purpose . T hey prevent by p as s in g of gas d ue to lar ge b ubbles by breakin g up the b ub bl e s , and re duce solid circulation , t hereby also r e d u c i n g the b ac k mixi n g of gas by moment of reactants adsorbed on the c a t aly st . In cases where one w ants to test w h et h e r t he flo w distributor op erates w ell , a no n a d s o rbi n g t racer is a more sensitive tool ( see Fi g . 19) . When there are b a ffl e s in a fluid bed , the problem becomes more complex . A good case can b e ma d e t ha t an ad s o r bi n g t racer is more s ensi ti ve , as it shows if a d s o r b e d tracer is recycled by the catalyst to the bottom of the T herefore , i f a baffled fluid bed ap p r oa ch e s plug fl o w wit h a r ea c to r . strongly a d so r bi n g rracer , it giv e s one better confidence in the d e si gn . However , one has to b e careful to c h ec k t hat the t r a ce r be ha v e s li n e a rly . To check t hi s , one alw ay s needs co m p a r ativ e result s from a p ack e d- b e d
1 29
Residence- and C o n tac t - Time Dis t ri b u tions
SF 6 has been reported to hav e desirable properties , and its ad reactor . sorption can be chan ged in a cold- flow model over a wide ran ge , as the adsorption equilibrium i s a function of the w ater content of t he air . If a properly baffled desi gn gives a residence- time distribution with a hi gh Peclet number ( low coefficient of variation ) for both a nonadsorbing and an adsorbin g tracer , thi s improves the confi dence in the scalability of the desi gn .
Un derstandi n g the effec t of adsorption on the residence - time distribu tion i s he lp f ul i n an ot he r w ay . I t explains why di ffe r en t reactions may have different sensitivity to scaleup in the same fl ui d bed . To understand this concept , one must first consider the evaluation of tracer experiments wit h adsorptive t racers in a more ri gorous way . T he approach u sed i s de sc ri b e d in detail in Shinnar et al . ( 197 2 ) . Consider a pack ed b e d or a chromato graphic column in w hic h a t racer can be adsorbed inside a porous p a rtic le . One can now perform a t racer experiment with a nonadsorbing tracer w hich can also diffuse into t he porous particles an d has t he same co effi ci en t of diffusion inside the particles . For simplicity a p ulse tracer ex periment will be used . T he density function of the nonadsorbing tracer i s desi gnated fo and that of the adsorbin g tracer f . Also a s s u me that the a adsorption equilibrium is linear and the amo unt of t racer adsorbed at equilibrium per unit volume of C o gas phase is dc 0 , w here Co is the con centration in the gas phase . Consider the case where ideal plug- flow con ditiond exist . T hen fo( t ) for t h e nonadsorbin g tracer is a delta function at t T. One can then measure fa ( t ) for different value s of T an d define a f unc t i on IJ.In ( t ) =
•
ljJ ( t ) a
=
f ( t - -c ) a
( 4 5a )
w hic h gi v e s the so jo u rn - ti m e distribution of the adsorbi n g t racer in the adsorbed state . The simplest case is if the tracer adsorb s in a way that the ra t e of adsorption and de sorp tion is very fas t . In this case ljJ ( t ) a
=
o
(en)
( 4 5b )
and for th e general case w here a is a cons tant , that gives the ratio of t h e residence time in the adsorbed p h a s e to that in t he contin uous p hase . ( 4 5c )
fini t e rat es , E q ( 4 5c ) will not hold . t he b e d has uniform prop erties and the flow in the gas phase is plug flow , lji ( t ) must b e a unique function of T in the continuou s ph a se . a One can f u r th er show that for vario us values of -c If ad sorp tion and desorp tion have
However , if
ljJ ( t / -c 1 a
+
• 2>
t
=
f
0
ljJ
ap
c e / , 1 ) \ji
ap
( ( 1 - e] / ;: ) d e 2
( 4 6a )
w here 1/Ja. p ( t / 1" 1 ) i s the function tfu p ( t ) given that T for the continuous phase is '1 · T he sub script p was added to in dicate that the flow in the continuou s phase is plu g flow . I f E q . ( 46) holds , one can al so show that the Laplace t ran sfer of IJ.Ia ( t ) must have the form ( S hin n ar et al . , 1 9 7 2 )
1 30
Shinnar
�ap and
(�)
e
=
- T P (s)
( 4 6b )
therefore ( 46c )
p ( s ) there fo re c haracterizes the adsorption process or in a more general w ay the transport processes in an outer p hase not reachable for the tracer use for f0 • In E q . ( 4 5c ) it was as s um ed that p ( s ) = as . If one as sumes a finite ra t e of adsorption an d desorption , then in the example given p ( s ) becomes P ( s)
a.s
A
=
7 1-+-=:-(a-:-: /}_,...,.)s
( 47a)
w here A i s a normali zed exchange coefficient . For understanding the gen eral concepts , one does not need more complex form s of p ( s ) . The re a de r should consult Shinnar et al . ( 19 7 2 ) for more details . If p ( s ) is given by E q . ( 47a) , then 1jJ A
ap
- e - CXTS / [ l+ ( CX/-A ) S )
( s)
( 47b )
_
is a well- known function used in chromato graphy . To d erive it , one must not only assume that fo( t ) is a plu g flow , but al so assume t ha t the catalyst bed has uniform properties and t hat t he c a t aly s t is s t ation ary T hi s mea n s that any molecule that got a d sorbe d returned to the same place it left . In the limitin g case w here A -+ oo [ E q . ( 4 5c ) ] , t he last assumption is not needed. H ow e v e r the assump tion of uniformity i s s till n ece ss ary If the bed is uniform in its properties and each particle returns to the same place in the gas phase , Eq . ( 46) can be generalized to the case where fo ( t ) is not plug flow . For any such c a se with arbitrary fo ( t) , which
.
,
$a C t )
t
ap
t
f (t)
a
and
$a ( s )
f 0 ( T ) 1J!
f0
=
f0
=
=
where p ( s )
f ( -r ) lji 0 ap
-t -r
(
d -r
t --' ,
( 48a)
)
( 48b )
d -r
r0 ( p ( s ) )
has
.
( 4 9a )
been substituted for the variable s in
f( s ) .
Si milarly ,
( 49b ) Note that in
the
previous simplified
case ,
1 31
Residenc e - a n d C o n tac t - T ime Dis t rib u tions
( 4 9c )
T he re s ults o f
Eq.
( 49c ) are experimentally testable , as one can me as u re
a s / ( 1 + as / A.) in a p acked- bed reactor under conditions approaching p l u g
Howe v er , t here are stron g re stri c tions on its applicability . E qu ation ( 49) also allow s computation of �a ( s ) an d therefore of 1/J ( t ) from measure a ments of fa ( t ) and f 0 ( t ) in o t h er cases w here the adsorbed particle returns to t he same place . T he only other case w here this direct c o mp utatio n is po ssi b le is w hen t here is n o correlation between the total residence time of a particle in the adsorb e d p hase and i t s residence time in t he ga s p h a s e . In this case flow .
( 50 ) w here * i s agai n t he convolution i n t e gral . Unfortunately , thi s i s a rather unre ali s tic case for a catalytic reactor . In real cases w hen can Eq . ( 49) b e use d ? One case is a packed bed with back diffusion . The other i s a stirred tank reactor in which bot h the continuous and gas p hase are mixed ( such as in a B erty reactor ) . I n that case all molecules in t he continuous phase have equivalent location an d the r e fore E q . ( 4 9 ) ho l ds . There are additional i m portan t i mplic a tion s that can be deduced from E qs . ( 48) and ( 4 9) . Consider the special case of a pa c k ed - b ed reactor in w hich the flow
o . Further assume that flow or r0 ( s ) = e all reactants and products have the same rate of ad sorption and d es o rp ti on and also the same adsorption equilibrium . T herefore , lJla ( t } is con stant for all compounds . Finally , assume t hat a firs t - order complex reaction oc cu r s In t hat case on e can w ri t e an equivalent of Eq . in the adsor�ed phase . ( 28) fo r the product distribution :
in the internal p h a s e
< c .> J
=
a.0 + J
n- 1
'E
r=1
-• s
is p l u g
1/J ( t.. ) ,..
a.
Jr
a
( 51 )
r
which for a packed - bed reactor can be w ritten
=
10
a.
n- 1 +
'E
( 5 1a )
r=l
In standard kinetic p r ac tic e one neglects adsorption processes an d simply assumes that t h e rea c to r is plug flow and o n e writes an equivalent of Eq . ( 2 6 ) :
P . < t > = c < t>
J
where /.. �
1
.
=
a. 0 + J
n- 1
E
* - T t.. r p
( 52)
r= l
are ei genvalues based on p henomenolo gical r at e constants . T is p time constant that has t h e di m en sions of time but does not refer to any real measurable time . It is exp ressed in lb fee d /lb catalyst per ho u r (or lb mo l feed /lb cataly st per hour ) . N ote t ha t
a
Sh.innar
1 32 1:
0
p( A ) r
=
1:
A*
( 5 3)
p r
Equation ( 5 3 ) implies t hat it is alw ays possible to describe a first -order re action occurring in t he adsorbed phase of a cat alyst by assumin g t h at t he reactor is a plu g- flow reactor . T he reaction rates will not relate to any real reaction rates at the catalyst surface , but this is not import ant in phenomenolo gical rate equations in reactor desi gn . Norm ally , tha kineticist is interested in eliminati n g from transport resistances that deal with transport to and from t he catalyst particles and di ffusion in the m acropores , as w ell as b ack di ffusion in the reactor . T hi s can be achieved by as suming that the cat alyst p articles are sm all enou gh and t hat the microreactor used is lon g enou gh [ Silverstein and S hinnar , 1 97 5 ) . I n thi s special case , one c an t hen estim ate t he conversion of an iso therm al packed - bed reactor i n w hich fo ( t ) deviate s from plug flow ( either due to di ffusional resistances or due to m aldi strib utions) , by meas uring f0 ( t ) with an inert tracer and then comp uti n g the effect of the deviation from p lu g flow j ust as one di d in a homo geneous reactor . T hi s is the re sult of derivin g E q . ( 5 3 ) from Eq . ( 5 1 ) . One must , how ever , note that fo( t ) has a different time scale t han E q . ( 5 3 ) . One m ust therefore non dimensionali ze f0 ( t ) and scale it such that the new first moment is the same as that u sed in comp uti n g T hi s is p articularly important , as kineticists often use di fferent definitions for ' P in comp uti n g Equation ( 4 9) to ( 53) also imply that for a first -order reaction , one can predict t he behavior of a p acked bed by getti n g kinetic measllrements in a completely mixed reactor such as a B erty reactor , and vice vers a . One cannot , how ever , predict t he outcome of any reactor for which E q . ( 4 9) does not hold , such as a fluid bed in which t he catalyst move s . If t he movement of the catalyst is slow comp ared to t he adsorption - desorption processes of all reactants , one can get approximate predictions . For exam ple , if the fluid bed is st a ged by vertical baffles , it approaches a series of stirred t anks or , for practical purposes , the react ants return to the same place . T he outcome c an then be predicted using a packe d - bed reactor ( or another fluid bed close to plug flow ) as a model . T he relationship between E q s . ( 5 1 ) and ( 5 2 ) w as derive d for the case w here 1Jia ( t ) is equal for all react ants . T his is not a very likely case but w as used for didactic reasons . How ever , the principle that one can pre dict t he outcome o f a first -order reaction in a reactor for w hich E q . ( 4 9) holds by me asurin g fo ( t ) and the overall kinetics in a packed - bed reactor is valid for arbitrary soj ourn -time distributions in t he adsorbed p hase , even when t hey are different for each compound . T he concept therefore has To prove t he validity of t he more general much bro ader applicability . principle , one can use an argument similar to t he one used earlier when w e discussed op tim al reactor concepts for nonisothermal reactors . For any first - order reaction one can derive a phenomenolo gical equation such as Eq . ( 5 2 ) . Kinetici sts define a first -order reaction by thi s condition as they do not measure real reaction rates in the adsorbed phas e . Consider such a case in w hich 1/Ja. ( t ) is different for all compoun ds . T he average sojourn - time di stribution of the molecule s will now depend on their st ate and therefore on the reactivity of the cat alyst . T hi s is included in :>.. * . Once again , t he hypothetical observer stands at t he outlet o f a plu g- flow reactor and collects the molecule s . T he molecules are now sorted into bins . T he molecules in each bin have a unique so journ time e in the continuous
:>.. ;
:>.. ; .
:>..; .
Residence- a n d C o n t ac t - Time Dis trib u t io ns
phase . I n each bi n , gi ven B one can comp ute P j ( B) , t he probability o f bein g in s t ate j given B. As the number of molecule s is large , p · ( B) i s eq ual t o Cj ( B) . Averagin g over all t h e histories yields < p or or the outlet c o n c en t r at ion . Note that if t he conditions leadin g to Eq . ( 49) hold [ nam ely , t he catalyst has uniform p rop e rti es and a molecule alw ays
?
1 33
�Ci>
re turns to the s ame place in the space of fo ( t ) ] , t hen is a unique func tion of the r esidenc e time t of the molecule in the no nad sorbed phase . I f one looks at molecules with different sojourn times i n t he no nadsor b e d phase , their pro b ability of being in state j should be the s ame as that of a molecule passi n g a plu g - flow reactor with resi dence time t in the nonadsorbed phase . The previous ar guments should t herefore hold re gardless of the fact that all compounds h ave different adsorption desorption behavior as long as the system is li ne ar and fulfills t he basic assumptions . T he p revious discussion leads to the di s co n c e rti n g conclusion that in order to predict the behavior o f a first - order reaction , one doe s not w ant a true residence -time distribution . A more accurate measure will be given T hus the choice of tracer by a no n ado s rb i n g tracer that measures fo( t ) . can be cruci al in p r e dic tin g the outcome even of a first - order r e act i on . There are c ases w here t he "true" residence - time distribution will not cor rectly predict t he outcome of even a first -order re ac tio n . T hi s has i m portant i m p lic ation s for t he scaleup o f first -order reactions . It was shown that t here i s a fun d a ment al similarity between a mixed reactor and a plug flow reactor and t h at one can alw ays scale up from one to t he other re gard less o f how the reactants behuve i n the adsorbed p has e . However , one cannot apply the kinetic data obt ained from eit her one to predict the outcome in a fluid - bed reactor . Thi s has important implications for t he scaleup o f first -order reactions , as it indicate s a fun d ament al similarity between a mixed reactor and a plu g - flow reactor in terms of arbi t r ary behavior in the adsorbed phase . An important exception is the limitin g case of a fluid bed which is rea son ab ly uniform and approximates plug flow . T his will occur when either ,\ is very la r ge or the mixi n g of the soli d is slow compared to t he tim e icale of l· ( t mixing > 1 /l_) . Ot he rwis e , 1/la ( t ) would depend on bot h l and solid mixin g . Unbaffled fluid beds in the b ubb li n g ran ge are not very uni form and are o ften modeled by a two - p hase model . Here , E q . ( 49) does not apply even if A i s lar ge . One can , however , use an ad so r bi n g tracer with a large value o f A to test how close the bed appro aches the desired condition , namely , that the bed has no bypass o f gas that does not contact the solid and ap proaches plug flow . Adsorbing tracers are an effective
tool to test w hether t he system approaches plug flow or not . In a packed- bed reactor one can measure .\ . In fact , if one uses tracers with low er values of A and still gets a-good approximation to plug flow , one can then get an es ti m ate of the solid backmixin g . To esti m ate its imp act on the reaction , one would have to bound A fo r all the re ac t ant s and products , which can in p rinciple be done by usin g tracer methods similar to those used in Fi g . 26. There is one essential point to remember in testi n g for maldistribution of flow or for the e ffectivenes s of internals in fluid beds and trickle beds . This is the need for a goo d comp arison case of a w ell - per for min g reactor . If the residence -time distribution obtained by a tracer experiment deviates stron gly from p lu g flow , it does not necessarily indicate a maldi stribution of flow due to b ad desi gn . It could be a fluid dynamic instability p henom e non which is common in fluid b e d s and which can also occur i n lar ge
1 34
Shinnar
trickle - bed re actor s .
In
trickle - bed
rea ct or s it can be due to di ffusion and
adsorption p henomena in t he liquid p hase . cannot di sti n gui s h
between
I nput output tracer experiment s
t he two cases .
W hat t hey can do is p rovide com
p ari sons bet w een two reactors and give clue s t o t he
tive
w hich is
m e asure s s uch as baffles ,
effectiveness
o f correc
a very valuab le tool p roperly used .
Mea s u rement of H o l d u p i n M u l t i phase S y s t e m s I n m ultiphase
If
systems
the sep arate holdup o f each p h a s e m ay be unknow n .
t he system i s si m p ly soli d gas or liq uid gas , t he holdup o f t he solid or
li q ui d m i ght be obt ained by a pressure difference . T he holdup of the gas is more di ffi cult to measure , as it m ay dissolve in t he liquid or adsorb on a solid . H e re tracer experiment s can be very help ful . One has to be careful wit h the choice of t he tr acer . For a solid tracer it is im po rt an t to find p articles that do not sep arate and have the same si ze distribution and den sity as t he solid s t u died . For best results t he solid s hould be impre gn ate d with t he tracer when possible . I f t he liquid doe s not adsorb or evaporate almost an y tracer will do , b ut i f it does , it i s best to tag the liquid or p art of it . C hoice of a proper tracer from t he gas phase presen t s similar prob lem s . I f t he gas adsor b s or is soluble in t he liquid , one needs a tracer wit h similar p ropert ies or preferably an isotop e of the real compound . T hi s When a suit able tracer is sometimes the only direct w ay to measure holdup . is available , two further requirements m ust be met .
1.
Accurate measurement of
total feed
rate
for the
flow
of the p hase
me asure d
Accurate measurement of the amount of tracer introduced
2.
In a
p u l se experiment
!0
Q C ( t ) is equal to
gral
oo
better than
one can use !0
t he holdup obt aine d
if the
the
always
amount of
care fully
tracer
check t hat the inte
introduced
wit h
an
acc ur acy
95%. In this case , one s hould never use Eq . ( 1 3) . T he first will then give an estim ate of t he holdup . In a step experi -
moment o f f( t ) ment
one should
tracer is
oo
is
( 1 - F)
dt
to
e stimate the holdup .
T he e s timate o f
includes adsorp tion p henomena . T herefore , of the real flow , compoun d , one has to be
one that
not an isotope
very c are ful in i t s interpretation . T he example given in Fi g .
It
i s based
on
dat a
supplied
1 0 c an
1 98 1 ) from a study of the E xxon Donor men t s here
were
problems . by E xxon ( Mills an d D udukovic , Solvent Reactor . T he tracer experi
serve to illustrate the se
t o the author
performed t o me asure holdup o f t he t hree p hase s .
T he
system . T he reactin g gas is a mixture o f hydro gen and hydroc arbons , and t he li qui d is a mixture of dis solved coal liquids and recycled solvent . T he solid p hase i s ash and coal particle s . The tracer used to measure gas holdup was an isotope o f ar gon w hic h dis solves in the coal liquid , and it is t hi s experiment t hat is shown in Fi g . 10 . The solu bility of the tracer at the reactor temperature an d pre s s ure can be me asured and , at equilibrium , the mole fraction of ar gon in the gas is five time s t he m ole fraction of ar go n in t he liqui d . In Fig . 10 t he Laplace transform of t he residence- time distribution com puted from the tracer response was plotte d . T he first moment can be ob t ained from it by estimatin g the tan gent of f( s ) as s + 0 . T his i s sy stem illustrated is a three - p hase
1 35
Residence- and C o n tact - Time Dis t ribu tions A
e as the flow is close to plug flow and therefore ln[ f( s) ] s line ( see Fig . l O b ) , the slope of whic h is to the first moment . The variance has to be es tim at ed from the second derivative near zero . This is not easy , but if the coefficient of variation is sm all , one can do it by model fit t i n g usin g relatively easy h re versus ( s ) i almost a straight equal ,
=
where
1 (1
+
T S /n )
( 1 9c )
n
This pp a several problems the fact cer sol What y the of the tracer , estimate l m r r e ce t lar there no that esti and the gas holdup . Fortunately , in this the es i t ev r reasons . Either gas-liquid mass r s or t the s small , to the has only very small im in the range of accurate measurements because it s The coefficient then in this is ap ec t then error the st the example ve , the mass e c e t could and this alternative could ruled It therefore a o e that mass transfer fast , hic that the the liquid phase the phase . the liquid h l u the of a o is No that if have to be difficult to e i at volume of the phase the reactor .
n in here is a c urve - fittin g p arameter . a ro ch was discussed earlier . are caused by that the gas tra is uble in the liquid . one reall wants to know is not holdup but an of t he gas vo u e in the eacto If t he co ffi i n of variation y is ge is w ay one can reliably particular case , y is small mate t m a e could be obtained . The sm all y could be due to s e al the t an fe r coe fficient is small i is If m as transfer coefficient is adsorp i n in very lar ge . liquid phase a p ac t on C 2 ( t ) , m ain ly affects C 2 ( t ) for t + co ( or + 0) . of variation is reality lar ge , but not p r ia e d when comp uted from the measured form o f f( s ) . One can get a lar ge 2 in e imate of ll and y . In gi n trans fer co ffi i n be measured indep endently be out . is re s n abl to assume the is w h allows one to assume gas adsorbed in is in equilibrium with gas I f one knows o d p and the solubility of gas , a condition for the e ffects b s rp t ion can no reliable way to be made . te y 2 was not very sm all , there make any such correction . and it would s t m e t he gas in
There
.
,
C O N T A C T -T I M E D I S T R I B U T I O N S
has in t n a sojourn-type in active phase ob by s tracer as d t s ds r tracer of i then , for u flow , can n y in the adsorbed phase
The term con tac t - t ime dis trib u tio n been used he literature in two O e is distribution t he which is tained u ing a sli ghtly adsorbin g well as a non a sorbin g inert tracer . I f one de si gn a e f( t ) of the a o bin g as fa ( t ) and the n na d sorb n g tracer as fo ( t ) , pure pl g one define de si t function of a sojourn - time distribution
ways .
o the as
w here < o is the first momen t of f0 ( t ) . This distribution was discussed in the prec e din g section , w here it w as shown t hat the function lj!a ( t ) can be measured not only for plug flow but also for a limited number of other cases . However , alt h ou gh this distribution is useful for reactor modelin g and for dia gn osing maldistributions , it is not directly useful for e sti matin g the behavior of a first -order reaction . The kinetic relations assume p lug flow in a packe d bed , whereas tracer experiments with a proper isotope tracer will not necessarily give plug flow
1 36
S ht n nar
( see Fig . 2 6 ) . A c lo se examin ation of Fi g . 2 6 not use the real residence- time distrib ution o f dict the behavior of a he te ro ge n eo u s reactor .
w il l demonstrate that one can the r e ac t a n t s t o re li ab ly pre In Fi g . 2 6 t he overall flow is plu g flow . However , the re sidence time of the 1 3 c tr ac er is not plug flow an d has a lar ge coe fficient o f v ari ation For ot h er reactants the dif fe re n c e could even be l ar ge r If one applies t he kin e tic rel ation s w hic h are b as e d on t h e assumption of p l u g flo w to a model b ase d on the act ual re si d en ce - t im e distribution , o n e would obtain very erroneous results . More me ani n gful results can be obtained by u si n g the data from t he h e li u m tracer . T he helium tracer has a re side nc e - t i m e distribution similar to fo ( t ) .
.
true overall residence-time distribution of any of the actual reactor is a fluid bed , one can no lon ge r rely on the re sults from a helium t r ac er , nor can one rely on the t ru e residence time distribution . However , if the true residence- time distribution for both the fluid bed and t h e packed be d are available an d t he y are very close to each other , one could deduce t hat the re ac tio n should scale up safely . In b u t dis similar to t he
If
re ac t an t s .
t he
other cases , one cannot directly apply the in formation . A m o re useful definition of contact - time distrib ution is b a s e d on t he re lation bet w een the product di s t ri b u t ion of first-order reac tio n s and t h e E arlier it w a s shown that knowl residence-time dis t ri b u ti o n of the s y s t e m edge of the p ro du ct distrib ution defines the Laplace tr an s for m o f the residence-time distribution density function [ Eq . ( 2 8 ) ] . F or a fi r s t or der irreversible re ac tion A -+ B with a re act io n rate r ( A ) = - ka , the outlet con c ent r ation in a hom o gen eo u s s y s te m all
.
-
"
a
-
a
==
( 5 4)
f( k )
f "
f{ k ) is t he Laplace tran s form of f( t ) w ith the reaction rate k sub stituted for s . For a first -order re act ion o cc ur rin g in a he t e ro ge n eo us re actor , one can use Eq . ( 28) to d e fin e the di s t rib u tion l!_ by w here
JP
==
a. 0 J
n-1 +
I;
r==1
a.
( 55 )
$c >.. r >
J r-
equivalent of Eq . ( 2 8 ) . This distrib ution , o ri gi n ally pro po s e d et al . , 1 96 2 ) an d Glasser ( Glasser et al . , 1 9 7 3 ) , is not a real distribution of sojourn times b ut has the s ame dimensions and p rop e r ties . Re gar d le ss of the rea� s oj o urn ti me distrib ution , A. r is defined and m e as ure d by a s s u m in g that .!J:( A.r ) is plug flow for a p acked - bed r e ac t or . For a complex first - order reaction o cc ur ri n g in a p ac k e d be d reactor , one can determine A. r for each r . I f the reaction then occurs in a different reactor s u ch as a fluid bed , one c an obtain me asurements of C j · The i n for m at i on from Cj an d the Ar fro m t he packe d - bed reactor giv�s estimates for several values o f j:( A r ) . T o ob t ain the complete fun ctio n j:( >.. r > , one would h ave to use cu r ve - fittin g functions . H owever , in p rac tic e this is o fte n not ne e d e d , as sufficient information is available from the linear v al ue s of _i{ A. r ) In this case , A r c orre s p ond s to the L ap l ace transform variable o f 1/J ( t ) even t h o u gh 1/J ( t ) is an artificial constraint defined by Eq . ( 5 5 ) with no physical me ani n g: Another w ay of m e asurin g iOr ) is t o vary k in Eq . ( 5 4) . In a cat aly t ic reactor , this can be d on e by eit h e r changing the te m p e r at ur e of the r eac to r or the re ac tivi ty of the c at aly s t . One can then use an equivalent of Eq . ( 5 4) which is an
by Orcutt ( O rc utt
-
-
"
•
137
Residenc e - and C o n t ac t - T ime Dis t rib u t ions
( 5 6) to define �.
I t h a s the same me anin g as the definition in Eq . ( 5 5) with k re placing B ut it may n o t have the identical numerical form . Chan ging temperature or reactivity m ay change the complex t ran sport proce ss as well a s the ad sorption and de sorption rate s in side t h e cataly st p article s . Therefore , the values of < k ) obtained in this w ay will not be identical to t hose obtained usin g W < A r ) as de fined in Eq . ( 55) . However , the form will be similar . As ume that one measures ( k ) or �( "- r > in a p acked bed and then re peats the s e experiments in a fluidized be d keepin g the sp ace velocity constant , and measures the conversion for each ( k ) . T he highest conversion possible at eac h ( k ) is the same as in a plu g - flow reactor . I f , for any ( k ) , the con version is higher [ or f(k) smaller] , it follows that either the reaction is not first order , or the reactor is not isotherm al , or t he initial small packed -bed reactor w as diffusion controlled . If the conversions for all values of ( k ) are equal to that of the packed bed , the fluid is essentially described by a p l u g flow . If not , the conversion at each point is a m e asure of the w ay the fluid bed deviates from the packed bed . If a /ao for the two cases is plotted vs . ( k ) , one can interpret the curves as plots of ( k ) for the two reactors . (k) for the p acked b e d is by definition plu g flow . Much c an be learned from the nature o f the plot . In Fi g 2 8 an experim ent al contact -time distribution for a relatively short bu bb ling fluid bed is ve n and compared with a plu g- flow re
t..;.
i
s
_t
_t
.
_i
gi
actor and a stirred t ank . Note that $ ( k ) of the fluid bed is lar ger than $(k ) o f the packed - bed reactor ( plu g- flow case ) for all values o f ( k ) . A t lar e values of ( k ) , < k ) , for t he fluid bed , is very flat and , within exp P.rime n tal accuracy , almost p arallel . This indicates that part of the material es sentially bypasses the reactor . In fact , a s im p le bypass model
g
i
i< k )
=
a exp ( - k T) + b
( 5 7)
fits the experimental result s quit e well . Fi gu re 28 also gives tank . Note that it fit s t he experimental data les s well . (k)
_t
$( k ) for a stirred is larger than that
1.0 .B
.6
t( k )
.4
.2 5
10
15
20
k
25
E x peri m e nt al cont act - time distribution for a flui d - bed ( Orcu tt
FIGURE 2 8
et al . , 1 9 6 2 ) . Comp arison with variou s models ; - - - - - , stirred - t ank plu g- flow re actor allowin g reactor ; - - - - - plu g - flow reactor ; for bypass and stagn ancy ( experimental) . ( F rom Glasser et al . , 1 9 7 3 . ) �(k) is measured here by conver sion of ozone in fluid be d over an iron catalyst .
k
is varied by dilatin g the catalyst with inert s .
1 38
Shin nar
of a s tirre d tank for sm all and l a r g e values of ( k ) . For intermediate range i< k ) is smaller than for a stirred t ank , w hi ch is typic al o f s o m e byp as s p h eno m en a . The $(k ) for the fluid bed shown i n F i g . 2 7 has anoth e r int er es t in g proper ty th at is common to most fluid beds . The s lo p e of $(k ) n e ar k -+ 0 is small e r than for a plug- flow reactor . ( The only w ay to k now this is to measure the reaction in a packed bed with the same cat aly s t under isothermal conditions . ) This indicates that not only does the bed h ave a bypass in the bubble phase , but that part of the c at aly s t p h as e is not w ell contacted . In the author's ex p e ri en c e properly baffled (or turbulent) fluid b e d s do no t s ho w any by p as s in the gas p h as e but will s h a re the p ro b l e m that the dense catalyst phase is not perfused . Such baffled beds be h ave like ne ar - p lug - flow reactors with a s m all e r e ffect ive cat aly s t volume ( about 50 to 70% of real c at alyst holdup) . ( A possible e x p lan at io n i s give n in Krambeck et al . ( 1 967) . ] I f one c o ul d re a lly m e a s ure t he slope of ji ( k ) for very small ( k ) , it would app roach T as k + 0. T his i s the s a me p ro b le m t h at w as i llu s t r at ed in Fig . 13 w hen m e a s urin g the tail of F(t) . I f a region of t he reactor i s n o t well perfused , T will be underestimated . If an i n d e p e n d e nt estim ate of T is available , underestimatin g T usin g t h i s technique is evidence that the reactor is n ot well p e r fu se d . For the case shown in Fig . 1 3 , one could estimate T from the volume of the reac t or and the fe ed rate . In the c a se of � ( k ) , the on ly way to e s ti m a t e the first moment is by usi n g a packed bed reactor for comparison . Note t hat $( k ) here yie lds some v al u ab le in formation directly ap p lic able to reactor de si gn . One can compare the infor m ation obtained from Fi g . 27 with that obtained by tracer exp erim en t s in similar fluid beds ( see Fi g . 1 7 ) . Fi gure 17 for a similar be d also clearly in dica t e s R by p ass phenomenon . T he e scap e probability A ( t ) m eas u re d from the tr ac er response of a nonadsorbing tracer show s a c le ar m aximum , indic at in g a b y p a s s or stagnancy . The method shown in Fi g . 17 has one significant a d v an t a ge . It a l lo w s one to p lot A ( t ) w hic h gives dire c t p hysical in s i gh t into the nature of the transport p roc e s s . T here is as yet no e q ui v alen t way to extract e qui valen t information directly fro m in s pe cti on of $ ( k ) . I f f( t ) is measu r ed , compu t a tion of f( s) or f'( k ) i s s t r ai gh t fo r w ar d an d relatively e asy . T he op p o si t e is not true . i:< k ) i s hard to in ver t accurately , a n d therefore it Js h ard to obtain an acc ur ate estimate of the escape p ro b a b ility A ( t) from ljl( k ) . There are way s of invertin g � ( k ) eit he r by n u m e ric al inverSion or by fittin g it wit h a theoretical mo del :- but bot h methods are inacc urate . Ho w ever , i: is
u niq u ely measurable only in
t he s e cases where the ab sorbed
in other word s , wh e r e ad s o rp t ion c han ge t he residence - time distribution in the continuous p h ase
tracer p articles return to the sam e pl ace , does
not
.
i&_( k ) provid es a useful alternative . Unfortunately , �( k ) s u ffe rs from t he same problem as lJia ( t ) . T h at is , although it is u s e fu l in mo d e lin g , it cannot b e used for scaleup , as the � ( k ) d e p e n ds on a d so rp tion behavior and would be di ffere n t for different fU'st - o rd e r reaction s . One must t herefore be careful w he n ap p lyin g $ ( k ) t o ac t u al s c aleup problem s . I t will , i n a c ol d flo w model , give good in dica tion s o f bypas s p he n om e n a . T here i s , ho w e v e r , no guar an t e e t h at t he As t hi s condition is oft e n not met ,
real re acj ant will h ave model . lji ( k ) is u s e ful trial reactor s .
one and
If
an
the same
behavior as reactions used in t he cold flow
in stu dyin g m aldistrib utions of flow in act ual indus
do e s no t pe r for m up to expect ations , � ( k ) by chan gin g the t e m p erat ure t he n c o m p are the effect of c h an gi n g in the
in d ustrial reactor
c an o ften obtain sever al va l u e s t h e r eb y
(k) .
One can
industrial reactor to
of
(k)
pi lo t plant re s ult s or similar indu strial reactors .
From
one can then infer � ( k ) and possible re a son s for the poor p er fo rm anc e . This is a power ful di a gno s t i c tool , complementin g or s u b s ti t u ti n g for tracer studie s in indu stri al reactor s . One can get similar re sult s even if the re ac tion is n on li n e ar and complex in volvin g several co m p o u n d s . One starts with the a s s u m pt ion of a plu g- flow re acto r , estimates a s e t o f rate const ant s , and t he n obtains e xpre ssio n s comp letely eq uivalent to E q s . ( 1 6 ) and ( 20) . this
In this c ase , unlike the homogeneo u s c a se , one
comp ute t he outlet composition .
cannot use f( t ) or lji ( t ) to -
o u tlet com p o si tion can still be used to e sti m at e $(k) , as � ( k ) is an operator t r an s for min g the results of a plug flow re ac t o r into our r e ac to r . Note t h at r eli ab le d e si gn requires that $ ( k ) be close to p lu g flo w . An especially i nt er e s tin g case is one of consecutive re ac tion s with equal
reaction r at e s .
T he
Consider , for example , t h e
ca se
The solution of this c ase is gi ve n in Table 2 . I f the values of the r e ac tion rates are equal , one gets an e s ti m ate n o t only of � ( k) b u t also of the hi gh e r derivatives of _i ( k ) or ( dn� ( k) /dkn ) k =k . i Even if the v alue s
of
( k ) are di ffe re n t ,
the values
of
C j give a goo d approximation of the are small . If esti mat es of the
derivative s as lon g a s the differences derivatives are availab le ,
expansion around ( k ) re actio n in
sec u ti ve
a
$(k)
can be recon structed from a
( see G la s s e t et
cold flow model .
al . ,
1 97 3) .
However , if
reaction , one c an comp are t he
It is
h ar d
T ay l o r
to
series
u se such a
an industrial reactor has a con distribution to that of a
p ro d u c t
If signi fic an t deviations are foun d , this allo w s so me nature of t he flow . A good b as e case , preferab ly an isot herm al packe d - be d reac t o r , is es s e n ti al as well as a well-per formin g re actor of a similar type . One doe s not al w ay s w an t to rec o n s t r u c t $ ( k ) ( alt h ou gh this is pos sible ) . T he nature and m a gnitu d e of t he deviations give some i mme di at e qualitative infor m ation ab o u t t he n at u r e an d m a gnit ud e of the deviation from plu g flo w , which c an b e i n t e r p r e t e d by looki n g at some sim p le flow models an d st y din g t heir effect on p ro d uct dis tri b ution . packe d - bed
conc l usio n s
r e ac to r .
as
to the
gi ve n in Table 3 . In t h e se examples , the overall resi dence time T is kept c on s t a n t for all c ases . The re ac t io n rat e k i s varied for each flow model so as to keep the outlet concentration of A ( A /Ao ) con stant . T hi s is don e to make t he e x amp le more consistent with act ual practic e . In m any c a se s , c atalyst activity c han ges and is c o m p e n s ate d for
An examp le is
1 40
Shinnar
TAB LE 2 Dep en denc e of t he Product Distribution of a Fir s t - Order Consecutive Reaction on the Residence - Time or Contact - Time
Distribution8
Reaction :
Reactor with residence time distrib ution f(t)
Plu g- flow reactor
A " A f(k 1 ) 0
-
)!_ A = k1 0
2
E ._1
1-
_Q_ = k k A 1 2 0
D
TI
e
'J-1' 2
- k, T
( k. - k . )
j :# 1
L
. _1 3 1- '
B A0
1
J
TI
J- '
j:# i
- =
1
-k.T
e 1 ( k . - k. )
'- 1 3
J
=
1
1
2
E
._1
1-
c
- =
A0
k k
1 2
f( k ) 1 ( k . - k. )
--::-:-=--,-
II
j =1 ' 2
J
j:#i
3
E ._1
1-
1
f(k ) 1 __.:;_ "
TI
j=1 ' 3
__
]
(k. .
j :# i
-
k.)
__
1
1-A-B -C
=1-A-B -C k.
]
a
k
For contac t - time di stribution , sub stitute
=
3
1
( k ) for
-
E
i= l
f(k ) 1
IT
j=1 , 3 j :# i
k.
k. - k . ]
1
f( k ) .
by changing the re ac to r temperature . In practice one would therefore comp are the results of the industrial reactor to th e pilot plant at constant I A o . One does the same if the conversion is reduced du e to a 'naldistribution i n the flow . H ere the c o n c e nt ration of variou s products of a consecutive re action , w hich are a function of t he residence-time di s t rib ut ion , can clearly show if the reduced conversion is due to a change in catalyst acti v ity or due to a m aldistribution in flow ( provided , of course , that one can confirm in the p ilot plant that catalyst reactivity does not strongly affect selectivity ) . The results for the case of co n s ec u tive reactions A -+ .... C -+ D w he r e all reac tion rates are equal are given in T able 3. The results in t hi s example are very sensitive to the residence-time and contact - time distributions .
A
B
1 41
Residence- and Con t ac t - T ime Dis t ribu tions
k 1 = k 2 = k3 = k Plu g - flow reactor e
-kT
Reactor with residence time distribution f( t )
AAo B
Ao
=
=
2
2
T �
k
e
Ao c
-kT
1 - A - B - C
1 -
f( k1) A
f( k ) = -k 1 ddk
=
I k=k
1
k 21 d 2f( k) 2! dk 2 k=k 1 1 - A - B - C
( if=O ( k.� ) i] e -k -r 1'
A P P L I C A T I O N S O F CO N T A C T - A N D R ES I D E N C E - T I M E
D I ST R I B U T I O N S T O T H E D E S I G N O F H E T E R OG E N E O U S R E AC T O R S
Earlier , rigorous concepts as to how residence - time distributions affect the
conversion and contact - time distribution in i sothermal homo geneous reactor s were discus se d . U n fortunately , very few reactions are truly homogeneous and fewer are isotherm al . In the previou s sections it was shown that t he se principles cannot be applied to hetero geneous reactors in strai ght forward way . However , the m ain applic ations are still valid if one is care ful to recogni ze t he special fe atures o f residence -time distributions in hetero geneous
...... ""' ""
TABLE 3 Depen dence of the Product Distribution of a First - Order Consecutive Reaction on Residence- Time ( Contact - Time) Distribution :
Numerical Example k
k
1
k
3
2 A --- B --e -D k
Flow model
1
= k
2
= k
3
= k
Average residence time ( h )
K
( h- 1 )
A JA
0
B /B O
C tC
0
D !D
0
1.
Plug flow
1. 0
1.0
0 . 368
0 . 368
0 . 1 84
0. 08
2.
Ten stirred tanks in series
1. 0
1 . 05
0 . 36 8
0 . 35
0. 183
0 . 099
1.0
1 . 06 8
0 . 368
0. 336
0 . 1 92
0 . 104
4.
1. 0
1 . 36
0. 368
0. 236
0 . 1 91
0 . 205
5.
1.0
4. 03
0. 368
0. 33
0 . 147
0 . 156
3.
1.0
2: CJ) ;:s ;:s Q ....
143
Residence - and Contact- Time Dis tri b u tions
reactors . I n this section t he concepts presented earlier w i ll be extended to the more general case o f hetero geneous reactors . I n the pre sent state o f knowle d ge , ri gorous al gorit h mic use o f residence time distributions to predict product distrib ution in heterogeneous reactors is very re st ric t e d , even for first -order reactions . T here is no real goo d algo rithmic equivalent of Zwieterin g' s method . Fortunately , t he c oncept of boundin g the outlet composition is still valid . U tili zin g different models , one can still evaluate w hether t he reaction performan c e is sensitive to the act ual flo w model . T he resi den c e - time and contact - time distributions can then be used to place c onstraints on the range of flow models that one should consider . If the re ac t or is close to plug flow , one c an evaluate bounds by me asurin g the deviation from plu g flow by m e an s of a tracer experiment . T hi s works well only if t he devia tion from plug flow is small . The c on c ep t u al uses of residence - time distributions given earlier are still valid . Plu g- flow reactor s or staged reactors approachin g plug flow are optimal for all first - order reaction s and for all reactions in which contact between product s and reactants should be minimi zed . In fact , the example mentioned earlier was from c atalytic heterogeneo us reactors ( B e le vi et al . , 198 1 ) . Also noteworthy is t he con c ep tual use of t he contact - time distribu tion � ( k ) in react or design and scaleup . Recall that ( k ) was really an oper ator that showe d quantitatively the w ay the trans po rt process in the reactor modi fy the p roduct distribution as comp ared to a p acked - be d reactor . Its time scale was the reaction rate used to pre dict the results of a pac ked bed reactor , w hich is simply the overall reaction rate p e r unit volume or per unit wei ght of catalyst . It therefore has the same time scale as the in verse of the space velocity . In a packed-bed reactor with a good flow distribution , it is possible to keep both ( k ) and fo ( t ) , the residence-time distrib ution in t he gas p hase , constant . H owever , durin g sc ale up , t hi s is no t automatically guaranteed . fo ( t ) will norm ally stay con stant and close to the plug flow if the microre · actor or p�ot plant is su ffici e nt ly lon g [ Silverstein and s hinnar , 1 97 5 ) . However , lj! ( k ) will stay constant only if there is no mass transfer resistance between t he gas p hase and the catalyst parti c le . Otherwise , increasin g re actor len gth will chan ge the mass tran s fer coefficient , which will chan ge ( t ) and t herefore (k) . Another interestin g con c ep t u a l conclu sion can be derived from the con cept of a contact - time distribution . I t can be seen from Eqs . ( 4 8 ) to ( 50) that , for certain classes of systems , tPa ( t ) and a ( s ) can be uniquely d e rived from two s e t s of information .
�
i
i
_i
�
1. 2.
T he residence - time distri b ution fo ( t ) in the continuous p hase T he residence -time distribution fa ( t ) of an adsorbin g tracer in a plu g - flow reac tor
Consider a first - order reaction that occurs in the adsorbed phase .
As
each of the different reactants has a di fferent adsorption behavior ( its adsorptive e q ui li b ri um may be different , as well as the rate at w hi c h it
adsorbs or desorbs ) , e ac h species has a different i ( t ) or (s) . To proper ly model the system , one would need an exact tracer (isotope) for each re actant . However , for t he special case described , one could predict lji ( t ) or fo r each of the spec i es , given t he tracer results from a plu g- flow reactor and fo ( t ) , which is common to all species . I f one follow s a molecule
_i
_i(s)
Shinnar
1 44
un der goin g a first -order reaction , one c an concep tually measure for each molecule a total s oj o u r n time i n the adsorbed p hase . To comp ute its residence - time distribution , one must also know the reaction rates in order to determine how much time it s p e n d s in e ac h of the various states it can assume . T his is si mi l ar to t he concept u sed earlier to deal with fi rst - order reactions in n o ni sot h e rmal homo geneous system T he above is equivalent to c han gi n g t he n at ure of p( s ) in E q . ( 4 9a ) . Al t ho u gh one can n ot make this computation without detailed knowledge of a d sor p tion kinetics , one can use the concept to derive an im p or t an cri terion in reaction engineerin g : n am e ly , that for all case s for which Eqs . ( 4 8 ) to ( 50) ap p l y , one can u se the p se udokinetics o b t ain ed from a packe d bed reac to r and ap p ly it to all case s , u sin g fo( t ) to comp ute the result s . T his allows one to d eal w i t h di s p ersion and recyclin g in p ac k e d beds . I t also show s that for a fi r s t - or der reaction , t h e result o f a stirre d - t ank catalytic reactor is direct ly scalable to a pac k e d - b e d reactor , and vice vers a , as a stirred tank is o n e of t he few cases wit h movin g c at al y s t for which Eq . ( 48) app li e s . T his s c a li n g i s po s silb e despite the fact that one is re ally m e as u ri n g a p s e udokin eti c first - order reaction rate b as e d on space ve lo ci ty and not the ac t ual kinetics based on r e al re sid ence t i mes . T his con c ept can be extended q u ali t at i ve l y to s ec on d - or d e r r e ac t i ons in a catalytic reactor . I n reactors o f thi s type , mic r o m i xi n g is no t impor tant . However , it is o ft e n important to know w hether t he p ro d uc t distribu ti on can b e p re di c t e d reasonalby from fo ( t ) . For cases w here Eq . ( 48) applies , the Berty re ac t or p rovides a simple test . If a s ti rr e d - t ank cat aly ti c r e ac tor y i e lds kinetic r el ation s that p redict well the behavior of a packed - bed reactor , t hen nonlinear interactions between products and react ants are not i m p or t ant an d fo ( t ) can be used to p r e di ct the effect of dispersion in scaleup . The comp arison above is often i mp or tant in the d e si gn of complex re It is t herefore important to understan d the p ri n cip le s by w hich actor s . E q s . ( 4 8 ) an d ( 4 9 ) w e r e derived . T he m ain assum p t i on s were : •
1. 2.
The system is sp atially uniform . A n adsorbed p a rtic le (or a p artic le diffu sin g i n t o a c at aly s t particle ) always r e tur n s to the s ame place [ or to a place equivalent to it in terms of fo( t ) 1 .
T he se assump tions are not alw ays valid . N everthele s s , t her e are many cases where one w an t s to predict the p er for m an ce of a hete rogeneous re actor w hich i s not a p acke d bed from eit her a s mall p acke d - b ed reactor or a p ilot plant . This applies to fluid beds as well as tran s p ort reactors such as a riser cracke r . I f o n e comp ares s u c h a reactor t o a p ac k e d bed , it is im p o s sible to keep b ot h fo ( t ) an d i_ 0(k) con s t an t . For example , the first I f t h e reaction occurs moment of f o ( t ) will be m u c h larger for the riser . only at the solid surface of this c atalyst , this s hould not matter . B ut in m any comp lex reaction systems c at alytic surface reactions compete and inter act with gas - p hase reactions . T he cat alyst or solid may also act a s an in hibitor for the g as - p hase reactions . In som e catalytic reactions , intermedi ates forme d at t he surface will either diffuse through the gas phase or re ac t in the gas p hase . One s hould therefore be careful in such cases , and evaluate the po s si b le impact of c h an ges in fo ( t ) relative to l/Ja ( t ) . Ri gorously , it is n ot enou gh that f o ( t ) and ljia ( t ) stay constant . T he distribution of sin gle stays in eit her p h a s e should also s t ay constant . I f
Residence- an d
Contac t - Time Distributions
1 45
one cannot assume this to be true , one can experimentally test t he sen sitivity of t he reaction s t o variation s in fo ( t ) , k eepi n g 1/!a ( t ) const ant . T his technique is a simple and power ful conceptual tool for or gani zing one ' s t hinkin g about the pit falls of a given scaleup problem . C onsider , for ex am p le , a catalytic fluid - bed cracker in which the crackin g really oc c u r s in the dilute phase riser . A t low t e m perat u re s , gas - p hase reactions are slow
and the res u l t s correlate very well with t ho se obtained in a dense - p hase bed at the same space ve loci ty . At hi gher temperatures , the gas -phase re actions become si gni ficant and the correlation is poorer . As t he se gas phase reactions are un d e si rab le , it is imp or t an t to reduce t he average residence time o f fo ( t ) , keeping the space velocity const an t . Anot her import ant example is t he c ase of a dens e - p h ase catalytic fluid bed . E q u ation s ( 48) and ( 49) do not apply ri gorously in a fluid bed for two r eas on s : 1. 2.
The gas phase is nonuniform , as part of the gas i s in bubbles and part in a dense phase around the catalyst . T he T he solid circulation and ad sorbe d reactants circulate w ith it . sojourn - time distribution at e ac h reactant depends on t he rate at which it adsorbs and desorb s from t he cataly s t .
One can , however , design a fluid bed s uch that Eq . ( 4 8) ap p ro xi mate l y For example , one can see to it that fo ( t ) ap pr o ac h es plu g flow . One can al so use an adsorbi n g t rac er with fast e x ch an ge an d desi gn the re actor suc h that fa ( t ) app ro aches plug flow . That would occur on ly if all gas particles are adsorbed and desorbed many ti mes . T hi s then ap p roac hes the condition of spatial uni for mi ty despite the pressure of bubbles . One can achie ve this either by w o r kin g in the tur bulent fluidi zation regime or by usin g properly desi gne d baffles . Designin g the tu rb u lent flow regime only help s condition 1 . The second condition c an b e approached only by b a ffli ng or stagin g t he be d . One can al w ay s approach plug flo w by a s u f ficient num ber of sta ges . I f the catalyst circulation between the st ages is slow comp ar e d to the rate o f a ds orp tion and desorption of reactants , one ap p ro ach e s condition 2 , na m e ly , that each molecule always returns to a place su c h that its r e side n c e time in t he continuous p has e is not c h an ged . These ar gu ments allow one to underst and on t h eo r etical grounds the B affl es give added confidence to t he scale importance of ba ffl es in scaleup . up even if the fluidi zation is t urb ule n t and a t rac e r experiment shows that T his is so b ec au se solid mixing one ap p ro ac h e s plug flow wit hou t baffle s . in a fluid bed alw ays incre ases d urin g scaleup even when t he pilot plant is reasonably l ar ge . It is i m port an t t hat t he b affles really reduce solid mixin g and divide the solid cat aly st phase into stage s . Vertical b a ffles , w hich are often used , do not achieve this goal and t h e refor e do not provide the needed safety net for scaleup which can be ac hieve d u si n g ho ri zon tal b a ffles . We will conclude with on e final e x ampl e . R e c entl y , fl ash hydro ge natio n of coal , in w hich coal i s devolatili zed and p art i ally gasi fi e d in t he p res e nce of hydrogen at hi gh pressure , has received consi d erable attention . A simplified kin etic scheme of the reaction is ap p li e s .
Coal
H2 H2 H2 - heavy liq ui ds - B T X -- - methane ( 1)
( 2)
( 3)
1 46
Shinnar
are also c at aly zed by the con sec utive reaction , plug flo w of t he gas is req uire d for good selectivity , and it is also important to control the re sid en ce time in a narrow optimal ran ge . Kinetic an d the rmo dy n am ic con si de r atio n s r equi r e t he reactor to be as isot herm al as po s si b l e It has and the e asi est w ay to achieve this is by backmixin g of solids . the re for e been su ggested that flui d - bed reactor s or risers with hi gh solid circulation would have si gni fic an t advantages for s uc h a de si gn . However , all pre sent experimental data were obt ained in c urrent dilute phase reactors in which t he solid / gas ratio is an order of m agnitude low er t h an would pre vail in a riser with solid recirculation . T here is no w ay to estimate reliably from data in a d il ut e p ha se cocurrent reactor w hat would happen in riser w it h hi gher concentration of solids . O ne cannot keep residence time and con t ac t time constant sim ultaneously . For any de si gn estimates one would need data fro m a system in w hic h both con ta c t time and residence time are similar to t he desired desi gn . Or one would need evidence th at one of these t wo has no i m p ac t on the re act io n T hese simp le examples should be sufficient to illustrate the varied ap p lica ti on s of contact - and resi dence - time di stributions to the desi gn of hete ro geneo us reactor . They provide guideline s for d e ali n g with th e diffi c ult proble m s of re ac t or scaleup i n complex reactors . Steps
2
and 3 can occur
coal ash as
w e ll as
by
in
t he
t he gas p h as e an d
char .
A s t hi s
is
a
,
-
.
N O T AT I O N
a J
A.
A(t)
concentration of compound A name of co mp o un d j inside the reactor fraction of artic le s inside t h e reactor that will le ave it be fore time t concentration of compound A in prod uct st re am fut ure life distrib ution of p article
p
C C
A Af
concentration of product f in feed stream
a
s trea m
C(t)
concentration of t r c er in exit
C (t) f
concentration of tracer in feed stream
E f f( s )
F(t)
expected value of t
n
n or J0"" t f( t ) dt
Laplace transform of f( t ) re si den c e - ti me distrib ution :
raction o f p artic le
f
enterin g at time
zero that leave t he reactor before time t
f( t )
g( t )
G ( t) k
kij
i i fr ac ti o n o f c s exitin g between t and t l1 t = f( t ) L1 t den sit y function of G (t) internal age distrib ution : fraction of p artic le s which are in t he reactor at a gi ven time of ob se rvation that have been inside the reactor less t han time t kinetic rate c on s t ant den s ty function o f re s d e n ce time distribution : p ar ti l e
rate constant of reaction A
+
i
,... A
j
147
Residence - and Contac t - Time Dis trib u t ions K
equilib rium constant of reaction A
+
Aj
i amount of tracer introduced in pulse experiment
ij
m
pi
probability of molecule or particle to be in state j
Q
volumetric flow r ate
t
time
Peclet n umber
Pe
V
volume of reactor variance of density function f( t )
G reek Lett e r s
constants in Eq . ( 2 5 )
�r
{V;
y = A
coefficient of variation of f( t )
T
ei genvalue s of firs t - order kinetic systems [ see Eq . ( 2 5) ]
r
escape prob ability o r inten sity function of residenc e - time distrib ution r E C t ) = rth moment of f( t ) r
A (t) ll T
r
=
average residence time in homo geneous reactor
=
V /Q
Laplace tran sform o f contact - time distribution sojourn - time distribution density function o f stay in the adsorbed p hase [ see Eq . ( 4 6 ) ] R E FE R E N C ES
Belevi , H . , J . R . Bourne , and P . Rys , C hemi c al selectivities dis guised by mass diffusion , Helv . C him . Acta , 64 , 1 6 1 8 ( 1 98 1 ) . Biloen , P . J . N . Helle , F . G . A . van der Berg , and W . M . H . S achfler , I sotopic transients in Fischer T rop sch reactions , J . C atal . , ( 1 9 8 3 , in press ) . D anckwerts , P . V . , C ontinuous flow systems , C hern . En g . S ci . , 2 , 1 ( 1953) . Evan gelista , J . J . , S . K at z , and R . S hinnar , T he effect of imperfect mix ing on stirred combustion reactor s , 1 2th Symp . ( Int . ) Combust . , Combu stion I nstitute , Pittsburg , P a . , ( 1 96 9a ) , p . 90 1 . Evan gelista , J . J . , R . Shinnar , and S . K at z , Scale - up criteria for stirred tank reactors , A I C hE J. , 1 5( 6 ) , 8 4 3 ( 1 96 9b ) . Froment , G . F . and K . B . Bischoff , C hemical Reac tor A nalysis and Design , Wiley , New York , ( 1 97 9) . Gilliland , E . R . and E . A . Mason , Gas mixin g in beds of fluidi zed solids , Ind . En g . C hern . , 4 4 , 2 1 8 ( 1 95 2 ) . Glasser , D . , S . K at z , and R . S hinnar , T he measurement and interpretation of cont act time distributions for catalytic reactor characteri zation , I n d . Eng. C hern . Fundam . , 1 2 , 1 6 5 ( 1 9 7 3 ) . ,
1 48
Shinnar
Hiester , N . K an d T . V erme ulen , S at u ration p e r fo r m ance of ion -e xchan ge and a d so r p ti o n columns , C hern . En g . P ro g . , 4 8 , 505 ( 1 9 5 2 ) . Himmelblau , B . M . an d K . B . B ischoff , Process A nalysis and Sim u latio n , Wiley , N e w Y ork ( 1 96 8 ) . K rambeck , F . J. , S . K at z , an d R . Shinnar , Stochastic models for fluidized beds , C hern . E n g . Sci . , 2 4 , 1 4 97 ( 1 9 6 7 ) .
S hinnar , Interpretation of tracer experi me n t s in sys te m s with fl uc t u a ti n g throughput , I n d . En g . C hern . Fundam . 8 , 4 3 1 ( 1 969) . Le hman , J . and K . Sch ii gerl , I n ve s ti g atio n of gas mixin g and gas di strib u tor pe r for m an c e in fluidi zed beds , C hern . Eng. J . , 1 5 , 91 ( 1 97 8) . L e ve n s p ie l , 0 . , and K . B . B i sc ho ff , Patterns of flow in chemical process vessels , Adv . C h ern . Eng . , 4, 95 ( 1 963) . Mills , P . L . , an d M . P . D u d u kovic , Evaluation o f Liquid - solid con t ac t i n g in t ri c k le b e d reactors by tr ac e r methods , AIChE J . , 2 7 , 8 9 3 ( 1 9 8 1 ) . M u rp hree E . V . , J . V oo rhies , Jr . , and F . X . Mayer , A pp lic atio n of cont ac t i n g studies to the an aly s is o f reac to r per form anc e , I nd . Eng. C hern . Process Des . Dev . , 3 , 3 8 1 ( 1 96 4 ) . N aor , P . and R . S hin n ar , Repr e sent atio n and evaluation of resi denc e time distrib ution s , I n d . En g . C hern . F un d am . , 2 , 2 7 8 ( 1 96 3 ) ; N aor , P . , R . S hinnar , and S . Kat z , I n dete r min ac y in the estimation of flowrate and transport functions from tracer e xp e ri m en t s in closed circ u l ati on , Int . J . En g . S ci . , 1 0 , 1 1 5 3 ( 1 9 7 2 ) . Naum an , E . B . , R e s i dence time distrib utions an d micromixing , C hern . E n g . C ommun . , 8 ( 5 3 ) ( 1 9 8 1 ) Nauman , E . B . a nd B . A . Buffham , Mixing in Co n t i n uo u s Flow Systems , Wile y , New Y or k ( 1 98 3 ) . N g , D . Y . C . , an d D . W . T . Ri p pi n The e ffe c t o f incomp lete mixin g on conversion in hom o ge n eo u s reactions , Proc . 3rd Eur . Symp . C hern . R eac t . En g . , P e r ga m on Pres s , Oxford ( 1 96 5 ) , pp . 1 6 1 - 16 5 . Orcutt , J . C . , J . F . D avi d s o n and R . L . Pi gfor d , Reaction time dis t rib u tion s in fluidi zed c ata ly tic reactors , A I C hE C hern . Eng. Pro g . S ym p . S e r . , 38 , 1 ( 1 96 2 ) . Orth , P . an d K . S c h ii ger l Distribution of re sidence times and cont act times in p acked bed re ac t or s : influence of t he c hemical reaction , C hern . E n g . S ci . , 2 7 , 4 97 ( 1 9 7 2 ) . Overcas hier , R . H . , D . B . T odd , and R . B . O lne y Some effects of baf fles on a fl ui di ze d system , A I C hE J. , 5, 5 4 ( 1 95 9) . 0 s t er gaar d , K . an d M . L . Mi c he lse n On the use of the i m p e r fe c t tracer p ulse method for determination of ho l d - up and axial m ixi n g , C an . J . C he rn . E n g . , 4 7 , 10 7 ( 1 96 9) . Pennick , J . E . W . Lee , a n d J . Maziuk , D e ve lo p m e n t of t he methanol to gasoline ( M T G ) P ro c e ss , 7th I nt . Meet . R eac t . E n g , Boston ( 1 982) . Petho , A . and R . D . N o b l e , eds . , Residence Time Dis trib u tio n T h eo ry i n C hemical E n gineeri n g , V e r l a g C h e m ie W ei n h eim , W e s t Germany ( 1 98 2 ) . S ei n fe l d , J . an d L . L ap i d u s Process Modeling, Es t imatio n , and Iden t ifica tio n , Prentice - H all , E n glewood C liffs , N . J . ( 1 9 7 4 ) . Shinnar , R . , On t he behavior of liquid di sp e rsi on s in mi xi n g vessels , J . Fluid Dyn . , 1 0 , Pt . 2 , 2 5 9 ( 1 9 6 1 ) . S hinnar , R . and P . Naor , R esi d en ce time di s t ri b utio n s in systems with i n t er nal reflu x , C he rn . E n g . S ci . , 22 , 1 3 6 9 ( 1 9 6 7 ) . Shinnar , R . , P . N aor , an d S . K at z , I nterpretation and e v al ua tio n of multi ple tracer exp e ri men t s , C hern . E n g . S ci . , 2 7 , 16 27 ( 1 97 2 ) . Krambec k , F . J . , S . Kat z , an d R .
-
,
.
,
,
,
,
,
,
.
,
,
149
Residence - and Contac t - T ime Dis t ri b u t ions S hi n n ar , R . , D .
Glasser , a n d S . K at z , First order kinetics in c o n tinuous
C h e rn . En g . S c i 28, 6 1 7 ( 1 973) . S hinn ar , R . , Tr a c e r ex pe rim e nts in re ac t or desi gn , 2nd C on f . Physico c he m . H y d ro d y n . , Oxford ( 1 9 7 7 ) . S hinn ar , R . P ro c e ss cont rol rese arc h : an evaluation of present st a t us and research needs , A C S Symp . Ser . , 72 , 1 ( 1 97 8 ) . Si lve rst ei n , J . , and R . S hinn ar , D e si gn of fixed bed cat alytic microreactors , Ind . En g . C hern . Proce s s D e s . Dev . , 1 4 , 1 2 7 ( 1 9 7 5 ) . Snow , A . I . , p riv ate comm unication , A R C O Petrole um C o . ( M ay 1 98 3 ) . T army , B . , private communication , Tracer experiments performed on the reac tor s ,
.
,
Exxon ED S Pilot Plant (March 1 98 3 ) . J . an d C . D . P r a te r S tructure an d analysis of complex chemical re act io ns , Adv . C atal . , 1 3 , ( 1 96 2 ) . Wein stein , H . and R . J . A dler , Mi c ro m i xi n g e ffects in continuous chemical reac to r s C hern . E n g . Sci . , 22 , 65 ( 1 96 7 ) . Wein stein , H . an d M . P . Dudukovic , Tracer methods in the c i rc u l at io n in Topics in T ransport P he n om e n a R . Gutfin ger , ed . , H e mis p he r e , New York ( 1 9 7 5 ) . Woodrow , P . T . , U se o f intensity function rep r e s e ntatio n of residence time variability to understand and impro ve p er for m anc e of industrial reactors , ACS Symp . S e r . , 6 5 , 57 1 ( 1 9 7 8 ) . Z virin , Y . an d R . Shinnar , Int . J . Multip hase Flow . , 2( 5 / 6 ) , 4 9 5 ( 1 97 5 ) . Z wieterin g , T . N . , The d e gr e e of mixin g in cont i n uo u s flow systems , C h e rn . E n g . S ci 1 1 , 1 ( 1 95 9 ) . Wei ,
,
,
,
,
.
,
3
Catalytic Surfaces and Catalyst Characterization Methods W . N I C H O LAS D E LG ASS
E D U A R D O E . WO L F
Purdue U nivers i ty ,
Wes t Lafay e t te ,
Univers i t y o f No t re Dam e ,
No t re Dame ,
In diana In diana
I N T RO D U C T I O N
C atalysi s i s a complex p henomenon i n which a multit ude o f i nt er e st in g
A review of the prin c i p l e s an d factors involve d in variables intervene . determinin g c a taly ti c ac t i vity is b e y o n d the scope of t h i s chapter . We have
focused on the characteri zation of cat aly t ic propert ies to enhance our un Emphasis is given to t he modern s u rface a n aly si s tec hniques , which can p ro vi de new in sights for interp ret ing catalyt ic phenomena . Such methods , whic h were often first applied to st u dy model surface s , are n ow b ei n g use d to analy ze the more complex sur faces foun d . on t ec hnical heterogeneous catalysts . T wo ty pes of c harac t e ri zation techniques are di scuss e d in this chapter : ( a ) p hy sical charac teriza tion , and (b ) physicochemic al c haracterization . T h e first type refers to total surface are a and p ore size distribution . Total area is re l evan t in de ter m inin g the contact between t h e catalytic a gent and reac t an ts . P oro s i ty is si gni fican t in c on t r o l li n g the tran sport of the c atalytic a gen t in to the support durin g catalyst p r ep ar ation as well as t he transport of r eac t ants and p r o duc t s between t he b ulk fluid p hase and the active sites durin g re acti on Since surface area and pore vol u me are r elat e d a b alance b etween these two variables is a key factor in cataly st de s i gn . The B ET a d s orp ti on method to me a sure total surface area a n d metho ds to m e a s u re pore s i ze dis trib ution are disc u s se d in t he follow in g two section s . O t h er imp o rtan t variables t hat cha r ac t e ri z e a su pp or te d cataly st are the area of the a c tiv e cataly tic component ( i . e . , metal area in t he case of a s up po r te d metal catalyst ) and the c ry s t alli t e size distribution of t he active Many reac tion s are sensi t ive to the crystallite size , and t hey are p hase . termed s t ruc t u re -sen s i tive or deman d i n g reac tions . Va ri o us factors are in volved in this structure sen siti v i ty The occu r re nce of a partic ular crystal phase , the ratio of edge to corner sites , the st r uctur e of the surface , and th e s tab ility of o v e rl ay e r s can all depend on c ry s t alli te si ze . A di s t in ct iv e chemical feature of c ryst all it e s in t he size re gion 0 . 5 to 5 nm is the low c oor dination n umber of the s urface atoms . Later s ec tion s de sc rib e sel ec tiv e de r s tan din g o f the underlyin g processes .
,
.
,
.
1 51
1 52
Delgass and Wo lf
chemisorption of gases as a method to measure total metal area an d x - ray diffraction an d tran s mission electron mic roscopy as methods to determine avera ge crys tallite si ze and size distrib ution . Although the methods mentioned previously have been known for many years , their development for chemic al analy sis of surfaces is more recent . As know ledge of the principles of op eration , capabilities , an d limitations of surfac e analysis methods expands , the in terpretation of catalytic p henomena will become accordin gly more sop hi sticated . Ultimately , thi s new knowledge of surface chemical behavior will support the search for and desi gn of new catalysts . Knowled ge of the composition of a surface is a prerequisite for understan din g its c hemistry . X - ray p hotoelectron spectroscopy ( X P S ) , Au ger electron spectroscopy ( A E S ) , an d ion scatterin g spectroscopy ( I S S ) p rovide quantitative analysis o f surfaces o r thin surface layers . This in formation can reveal surface en richmen t in alloy s , the surface concentration of promoters in multicomponent supported c atalyst s , and the composition of the s urface of mixed oxides . These methods can also detect imp urities and poison s an d , in general , follow c han ges in the cataly s t surface after expo sure to the reaction environ ment . Oxidation state an d p hase iden tification come most readily from XPS an d , for selec ted elements , from t he Mossbauer effect . T he c urren t frontier in surface analysis is the meas urement of local s urface composition an d structure by application of ( a ) extended x -ray ab sorption of fine struc ture ( E X AF S ) , an d (b ) secon dary ion mas s spectros copy ( S IM S ) . We disc uss the se analy tic al tools and magnetic resonance in relation to the determination of surface composition an d structure . T he distrib ution of a catalytic material , or a poison within ·a support , can be a key variable in c atalyst performance . Slight chan ges in p rocedure durin g cataly st preparation may in troduce significant c han ges in t he distri b u tion of cataly tic materials inside the pellets . We disc us s the use of elec tron microprobes and electron microscopes equipped with analytical capabili ties to assess the distrib ution of materials in side p ellets . To complete the chemical c haracteri zation of a surface , one must examine its interac tion with adsorbin g an d reac tin g gases . We discuss methods by whic h the effec ts of gas - solid contac t on the composition an d chemistry of the soli d surface can be examined . T he influence of the surface on the adsorbed molec ules is usually probed with vibrational spec troscopy ( I R or Raman ) an d can be investigated with a magnetic resonance ( N M R or E S R ) . Another measure of gas - surface interac tion is the stren gth of the adsorbate bon d . Sabatier suggested lon g a go that this bon d must be stron g enou gh to perturb the molec ule b u t not so stron g as to form an un reac tive s urface compoun d . T emperature- p rogramme d desorption and reaction methods for studyin g surface bon ding are discusse d , alon g with vibrational spec troscopic tec hnique s for examinin g the adsorbate molecules themselve s . T hese adsor bate studies can serve either as a direct probe of the reaction it self or as an additional mean s of chemical c harac terization of the catalytic surface . T he tec hniques revie wed in this c hapter are those best developed and most commonly applie d or most promisin g at this time . T he presentation emphasi zes the es sential conc epts an d features of eac h technique and major application s in cataly sis . We have attempted to avoid excessive use of jargon to keep the disc ussion s un derstan dable to the non specialist . Refer ences to more detailed descriptions are given in each section . We note , finally , that thoro u gh characteri zation of catalysts often requires use of combination s of techniques chosen to give complementary information , and
1 53
Cataly tic S urfaces
that characteri zation o f the catalyst surface encompasses only half of the information neces s ary to fu lly un derstand a catalytic system . T he other half is the evaluation of t he element ary kinetic steps of the reaction w hich are the m anifestations of the influence of the c ataly s t surface on the react
in g molecules .
TO T A L S U R FA C E A R E A :
BET METHOD
There are several technique s t o e sti m at e the tot al surface area an d pore size distribution of porous m aterials . Various review article s concerning such measuremen t s have been published ( E m met t , 195 4 ; Innes , 1 968) . The B E T method is , however , the most com mon t e ch niq u e employed today for such measurements ; conse q uent ly , it is the only t e chnique di scussed here . The reader is directed elsewhere in reference to other techniques ( I nn e s , 1 96 8) . BE T A dsorp tio n
Many physical adsorption isotherms exhibit S shapes which are inconsistent with the Lan gmuir i sot herm b ased on monolayer adsorption - de sorption e quilibri u m . B runaue r , Em mett , and Teller ( 1 93 8) p roposed an explanation base d on the assump tion that m ulti l ayer adsorption could occ ur . T he B E T
model also pos tulates s ur fa ce h omo gen ei ty and no intermolecular interaction The most common form of the B E T a ds ortp ion isotherm also assumes that an infinite n umber o f multilayers can be ad sorbed . T he resu l tin g lineari zed B E T i sotherm has the form
between adsorbed s p e ci e s .
V d (P a s
p
O
·
- P)
=
1
V C m
+
c - 1 V C m
p P 0
( 1)
where Vads is the volume at STP occupied by molecules adsorbed at a pres sure , P , Vm is the volume corre spon din g to monolayer coverage , C i s a constant , and P o is t he saturation vapor pressure of t he adsorbate over a plane surface . A plot of P / V ads ( P o - P ) versus P /P o is a s t r ai gh t line with slope S = ( C - 1) /Vm C and i n t erc ep t I = 1 /V m C . Knowin g S and I p e rmits c alc ulation of V m and t here fore the number of gas molecules ad sorbed in a mono l ayer , w hich , when m ultiplied by the cross- section al area of the ad sorb ate , give s the total surface are a o f the solid . T he followin g cross - sectional areas (in square angstroms per mole cule ) give reasonably self-consistent total are as : N2 16. 2 , 02 1 4 . 1 , Ar = 1 3 . 8 , Kr 19. 4 ( Adamson , 1 97 6) . N 2 is the p re ferre d adsorbate , but Ar and Kr are also commonly used . =
=
=
T he most common e xperimental con fi guration s used to measure B E T iso therms ( Innes , 1 96 8 ; E vere t t and Otterwill , 1 97 0 ) are ( a ) the classic al
volumetric static app aratus used by E m m e t t , in w hich pressure associated with increm en t al volumes of gas is measured ; ( b ) a recirc ul ati n g batch flow system with helium dilutent ; and ( c ) a pulse chrom atographic dyn amic
flow system .
Several commercial s y s te m s b ase d on s t atic or d y n amic techniques are available ( Everett and Ott er will , 1 97 0 ) . T he expe ri m e nt al obje c ti ve in every These case is to determine V ads as a func tio n of the adsorb ate pressure . values are then sub stituted into Eq . ( 1 ) so that the intercept and the
1 54
Delgass and Wo lf
slope of the lineari zed BET isotherm c an be determined . T he re gion of best fit for the isotherm is in the P /P o r an ge 0. 0 5 to 0. 3 . C ontinuous an d autom atic systems have also been de si gned and differ from batch systems in that the gas is added continuously to the system at a very low rate instead of in lar ge increment s . Contin uous a d dit ion per m i t s automation , w hich , when coupled with microcomp uters , provides di rec t c alculation of surface area an d pore si ze distribution . M ajor factor s in all B E T measurements are p roper de gassin g of the samp le and e q uili bration . O t her t e c hni q ues used for c alc ul ation of total surface area , such as gravi metric techniques , se le c tive a d so r p t ion from liquids , and small - an gle x - ray scatterin g , are briefly summarized by I nnes ( 1 96 8 ) . PO R E S l Z E A N D D I S T R I B U T I O N
T he distribution o f pore si zes i s one o f the i m p o rt an t characteristics o f s up ported c atalysts since i t i s related to the value of the effective transport coefficients , and consequently it can affect activity , selectivity , and rates of de activation . C at alyst pellet s have a comple x p ore structure which pre sents a wide distribution of si zes . Pores b et ween 10 and 1 0 0 A are re ferred to a s mtcropores , those between 1 0 0 0 A and 10 �m are called mac ro pores , and those of intermediate size are c alled mesopores . T he micropores are us ua lly characteristic of the s upport p o ros i t y , whereas macropores can ori ginate in the interp ar ticle s p ac e crea t e d d urin g formation of the pellets . A m ateri al e x hi bi t in g both micropores and macropores is described as having a bimodal pore di s t ri b u t io n . Detailed disc us sions of pore structure and its determin ation have been presented in various textbooks and mo no grap h s ( E verett and O t terwill , 1 97 0 ; Gregg an d S i n gh , 1 96 7 ; Parfitt and Sin gh , 1 976) .
A first - order approximation or estimate of the ave r a ge p o re si ze can be pe r fo rm ed by ass umin g that all pores are nonintersectin g c ylin de r s of uniform len gth L an d radius rp ; t he n
Pore s u r fac e Pore volume
=
s
v
p
=
2
r
( 2)
p
or
2V
r
p
___E. s
w here V p is t he t ot a l pore volume and S the total surface area . For materials with a distribution of pore si zes , it is necessary to kno w
the pore volume at various pore si zes . The derivative of the pore volume curve wit h respect to the radius give s the pore si ze distribution . T he two m os t common tec hniques used to determine pore volumes are the BJH method ( B ar re t t et al . , 1 9 5 1 ) , base d on t he use of either a d so rption iso therms or porosimetry . Pore Size Dis tribu tion from Phy s ical A ds o r'p t i o n Isother'ms
If a physical adsorption isotherm is e x t e nd e d to the re gion in which P /Po about 1 , a rapid increase of V occurs due to condens ation of the ads
is
1 55
C a taly tic S urfaces
ad s o rb at e o n ! he pore walls . Conversely , a de c r e a s e in pressure res ults in e vap o ration of the adsorbed o r condensed liquid . Kelvin derived t he followin g relation between the vapor pressure reduction over a liquid con tained in a c y lind rical c api llar y and the radius r : =
where P o
2V o
rRT
cos ¢
( 3)
the sat u r ation pressure , a t he surface tension o f t he liq ui d , simplicity ass um ed to be 0° ) , and V the molar volume of the liquid . T he K elvin equatio n c an be u s e d in pri n ciple to calculate r fo r any c orresp on din g value of P /P o . However , sever al com plic at ions arise : ( a ) a hysteresis loop is often observed i n ad sorp t ion isotherms ; ( b ) the pores are not cylindric al and have v aryin g radii ; an d ( c ) an adsorbate film of varyin g t hickne s s decreases the e ffec tive Kelvin radius . A disc u ssion o f the various theories that have been p ro p o s e d to account for the factors cited above is given by G r e g g a n d S i n gh ( 1 967 ) . A detailed c alculation p rocedure , known as the B J H me t ho d , has been reported by B arrett , Joyn er , and H alenda ( 1 95 1 ) . The m e t hod c on sists o f step w i s e c alculation s based on t he dat a obtained from the de s orp tio n branch of the is other m abo ut the re gion where P /P o :::: 1 . T he c alc ula tio n s yield a cum ulati ve pore volume versus po re radius whic h is then differentiated with respect to the radius to obtain t he pore si ze distribution . is
( 5)
22 , 400 m
adsor ption stoichiometry ( s urface atom s /n um ber of adsorbed , V ads the volume of gas adsorbed at monolayer coverage and standard temperature p ressure , and m the mas s of catalyst used .
w here v is the molecules
TAB LE 1
S elective
C hemisorption
of G ase s
on Metals
a
G ases Metals
Ti , Z r , H f , V , N b , T a , C r , Mo , w , Fe , Ru , O s Ni , C o Rh , P d ,
Pt , I r
02
C H 2 2
C 2H 4
co
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
Mn , C u
+
AI , A u
+
+
Li , N a , K
+
+
M g , A g , Zn , C d ,
+
I n , Si , G e , S n , Pb , A s , S b , Bi
a + , S tron g chemi sorption occur s ; ± tion is unob servable .
c hemisorption
H
is weak ;
2
C02
N
2
+
+
±
c he mi s o r p -
1 57
Cataly tic Surfaces T he metal fraction expose d , or dispersion ,
D
::
metal surface atom s tot al met al atom s
=
V lJ
.M
1
--
mw
X
10
-6
is c alc ul ated as ( 6)
where M is t he molecular wei ght of t h e metal , w t h e fr ac ti on al metal load up t a ke in micromoles . T he average c ry stal lit e si ze is c al culated by assumin g that crystallites are of t he same r e gul a r uniform s h ape ( i . e . , crystallite s are often ass umed to b e c ube s with five faces ex posed and the sixth bein g in contact wit h the support ) . For this
in g , an d ll i t he
geometry d
( 7)
wher e P M is the density and d the ave r ag e c r ys t alli t e s i z e . Spenadel and B o u d ar t ( 1 96 0 ) de m on st r at ed that H 2 c h e mi s orptio n on Pt black ( 10 torr , 2 5° C ) a gree d well with B E T d at a . A greement b et w ee n H 2 c h e m i sorp t ion and x - ray d i ffraction re sults was also obtained in t he case of a Pt supported on alumin a c ata ly s t . A variation of t he direct che m i so rp t ion techniq ue is titration of pre adsorbed 0 2 w i th H2 ( Gruber , 1 96 2 ) . B en son and B oudart ( 1 9 6 5 ) propose d that the ti t ra ti o n stoichiom e try is
w hic h allows for the up t ake of three h ydro ge n atom s per Pt at om , thus in t he· se n si tivit y of t he techniq ue . Mears and H ansford ( 1 9 6 7 ) found that a di ffe ren t Pt - H 2 stoichiometry appeared to be ap plicable to their result s . The dile mma w as resolve d by Wilson and H all ( 1 97 0 ) , w ho found that the s to i c hio m et r y of oxy gen chemisorption d ep en d e d on c r y s t al lite si ze . For this reason , titration t e c hn i q u e s are not used wit h s am p le s havin g very small si zes . T he u se of CO c he m isorpt io n has also been pro p osed to d e t e r m i n e H ow ever , the relative c on t ri b u metal surface areas ( H u ghe s et al . , 1 96 2 ) . tion of t he linear and br i d ge d form s of CO m u st first be assessed to e s tablish t h e proper stoichiometry . O t her gases , such as NO , have been used to evaluate the metal surface are a of R u ( R amamoorthy and Go n z ale z , c re asin g
1 97 9 ) .
The experimental te ch niqu e s to meas ure V a ds have not been mentioned above , but in m o st references a st atic or volumetric te c hni q ue similar to the one use d in BET ab s orp tion has been used . A flow pulse tech ni uq e t ha t p er mi t s rapid determination of V ad s has been described by several authors ( Gruber , 1 96 2 ; B en e si et al . , 1 97 1 ; H uas en and Gr ube r , 1 9 7 1 ) . T he effect of s am p le volume , amount of catalyst , pretreatment , an d so on , on t he value obtained for V ads usin g t he pulse technique has been rep ort e d . P ulse results agree well with the static te chnique s if ad so rption is very fas t and irreve rsib le . Reversible ad s orp t io n introd uces tailin g on t h e emer gi n g peaks , which can lead to error s ( W anke et al . , 1 9 7 9 ) . A s t rin ge n t re q uirem en t of the pulse t echni q ue is the use of clean
1 58
D e l gass and Wo lf
ultrahi gh - p urity gases and various trap s to eliminate trace imp uritie s . S arkany and Gon zale z ( 1 982b ) have discusse d conditions for proper use of the p ulse technique , in p artic ular , linearity of t he T C cell respon se and t he effect of reversible a dsor p tion Approp ri at e ly used , the flow technique gives good and expedient re sult s and the app aratus is easy to build . For a new catalyst it is advisable first to establish the operatin g conditions in w hi c h reproducibility with the static or other techniq ue s is attained . .
D E T E RM I N A T I O N O F AV E R AG E C R Y ST A L L I T E S I Z E A N D C R Y S T A L L I T E S I Z E D I ST R I B U T I O N X - ray D i ffract ion X ray s have often been used in c at alytic work in relation to the e stimation of cryst allite si ze and to obtain information on t he b ulk crystallograp hic st ructu re of catalytic m ate ri al s A n u m ber of mono grap hs exist on x - r ay diffraction ( Klug an d Alexander , 1 97 4 ; Cohen an d Schwart z , 1 97 7 ; C ullity , 1 9 7 8 ; Schult z , 1 98 2 ) and the subject is so exten sive t hat it can be treated here only in basic ter m s .
.
N ature of X rays X r ay s are electromagnetic r adi at io n with w avelen gths in the an g str om ran ge . Con sequently , they can penetrate m atter and therefore are especially well suited to probe the structure of solids . X rays are c om monly gen e r at ed by bom bardin g a solid ( tar ge t ) with hi gh- energy e le c tr on s to create inner - shell electron vac ancie s . T w o types of x rays are thus generated : ( a ) a c on tin uo us spectrum , and ( b ) a characteristic line spectrum . T he con ti n u ous spectrum arises from t he deceleration of t he incoming hi gh- speed e le ct ro n s by t h e tar get . Li n e s pectra occur when an electron from a high-energy orbital fi lls a low - ener gy v ac anc y in th e inner electron orbitals . The line spectrum is characteristic of the emitting m at eri al an d use used for x -ray spectroscopy , w hich will be dis c us se d in detail later . The p eaks of t h e line spectra are desi gnated by the orbital ( shell ) into w hich the electrons fall ( K , L , M , N ) and a Greek letter indicatin g the ori ginal orbital ( a , S , etc ) . Due to th e characteristic well - defined ener gy , the x - ray lines ar e us e d as a quasimonochromatic x - ray source in di ffraction
equipment .
X -ray D i ffrac t io n
Diffraction occurs when a w ave interfere s with an array of sc atterin g cen ters , causin g the ou tc o m in g wave s to reinforce e ach other ( constructive interference ) or to be out of phase and cancel e ac h other ( de structive inter ferenc e ) ( see Fi g . 1a ) X rays are scattered by the electrons of the irradiate d m atter . Elastically sc a tt ere d x rays have t he same freq uency as the inco min g x ray s ( coherent scat t eri n g ) ; t he opposite is true for in elastic scatte ri n g ( incoherent scatterin g ) . Elast i c ally scattered x rays , which are imp or t an t in di ffr action can be depicted as reflected by the scat t e ri n g ato m . Fi g B ragg used the reflection an alogy to explain x -ray di ffr action u r e 1b s ho w s a mon oc hro m atic x - ray beam of w avelength A directed on t wo succes sive planes of a crystal . T he reflected beam ( i . e . , scattered ) •
,
.
1 59
C ataly tic S urfaces
SOURCE
(a)
SOURCE
CONSTRUCT I V E
DES T R U C T I V E
(b) FIGURE 1
Schematic representation o f ( a) constructive (in p hase ) and de structive ( out of phase ) inter ference w hen a w ave motion i s scattered by slits , and ( b ) B ragg' s reflection analogy for x - ray diffraction showin g con structive interference .
and associated wavelen gths are also shown .
I f the reflected r ays are to be in p hase , the path di fference of the two successive incomin g rays , CB + BD = 2d si n a , m ust be equal to an inte gral n u m be r of w avele n gt h s , that is , 2d sin 9
=
N >.
The equation above , known as B ra gg ' s law ,
( 8) relates the d s p acin g o f the
crystal wit h t he an gle of inci dence an d wavelen gt h o f the i nco min g x rays . N is known as t he order of the reflection . E quation ( 8 ) indicate s t hat measurin g the intensity of the di ffracted beam as a function of the incident an gle gives a diffraction p attern which is characteristic of the crystallo grap hic str ucture of the irradiated s ample . S uc h measurement s , w hic h con stitute o n e of t h e m any m e t hods of X R D analysis , are most easily at tained with an x - ray diffractometer .
1 60
Delgass and Wo lf
An x - ray diffractometer con sists of a cir c u l ar table with a stationary x - ray source an d a movin g detector locate d in t he circ umference of t he t a b le with t he sample set at the center . T he moving detector , u s ually a p roportion al counter , records t he inten sity of t he reflected beam as a
function of the reflected an gle 2 9 . A strai ghtforwar d app lication of XRD is t he u se of the X RD p attern s to identify t he various phases existin g in a m aterial . Diffraction patterns of powders are compi l ed in t he Pow der Diffraction Fi le Searc h Manual ( B erry , 1 97 5 ) .
C rys tallite Size D e terminatio n fro m XRD Line B roadening Diffraction line s should , in p rinciple , be very narro w ; however , when the
size of a polyc ry st alline material falls below 1000 A broadeni n g of t h e diffraction lines is observed . I n strumental limitations and lattice strain c an also c ause line broadenin g . Line broadenin g due t o p artic le si ze only arise s bec ause of incomplete de str uctive interfe rence . When the incident be am is slightly o ff the B ragg an gle , the reflected ray which would nullify it ori ginates from N p lane s in side the c rystal . I f the crystal is sm aller than about 1 0 0 0 A most crystalli t e
,
,
planes do not have their destr uctive counterparts N p lanes aw ay , so that finite diffracted lines will be ob served at t he off- B r a g g an gle se t tin g ,
Various measures o f the peak broadenin g are causin g line bro adenin g . u sed , the most common bein g t he width at half peak i n t e n sit y 8 , and t he ,
I d 9 /I w here I is the ob serve d inten sity 13 . fe 1 max of the diffrac ted bearJ . T he relation between line b ro a d eni n g and crystallite size for a stre ss free materi al , known as t he Scherrer form u l a , is
inte grate d breadth ,
=
K A.
8
hkl
cos
.,.
92
e
( 9)
where K is a const an t that depends on the de finition of 8 used and the crystal geometry . T he con stan t K v arie s from 0 . 98 to 1 . 3 9 , b ut because of e xp erimental uncertaintie s , the con stant i s o ften set equal to 1 . 13 is de fined as t he inte gr al breadth ,
W hen is a volume aver age crystallite
si ze .
In the ab sence o f lattic e stre s s , b r o a denin g also occurs due to experi mental limit ation s such as nonpar allel and nonmonochro m atic irradi ation .
Other experimental factors also contrib ute to b roadenin g , even if the
s p eci men is of "in finite si ze" (i . e . , l a rg er than 1 0 0 0 A ) . C onseq uently , for t he correct e s ti m at io n of cryst allite si ze u sin g Scherrer , s form ula , the experimental contri bution to line broadenin g must be separ ated to obtain the pure diffr action broadenin g . Alth o ugh various procedures exist , two are o utlined here :
(a) a simp lified p rocedure , an d ( b ) an accurate
Fourier tran s form met hod .
Simplified Procedure :
In this c ase t he sample is mixed wit h a standard
that has p ar ticle si zes greater t han 1000
A and p roduces a di ffraction line
near the line o f the sample un der analy si s .
Altern atively , di ffraction p at
tern s o f t he samp le and o f another sample o f the same m ateri al containin g cryst allites lar ger than 1 0 0 0
A c an
be u se d .
In either c ase , if B is t he
width of t he samp le containi n g sm all crystallites an d b the width of the
1 61
Cataly tic Surfaces stan dard s amp le wit h lar ge cryst allites ( > 1000
f3
in g of the pure diffraction p a ttern
A),
the breadt h or bro ade n
i s e s tim ated by
( 10) or
S
=
B - b
( 1 1)
T he first correction i s d eri ved by ass umin g that t he shape of t h e peak is gau ssian , whereas t he secon d one corre spon d s to a C auchy p rofile .
B would be either
13t
o r Bi .
H ere
O ver t he years , the gau ssi an approximation
has been wi dely use d , altho u gh some evidence indicates that the p ure dif fraction p rofile is better ap pro xi m ated by the C auchy fo rm when a di stribu tion of p article si ze exist s ( K lug an d Alexan der , correction c an be used .
1 97 4 ) .
H owever , either
I t must be emp hasi zed that because the shape of
the pro file s m u st be assumed , the simplified procedures gi ve speedily re sults of good relative accurac y b ut have restricted ab solute signi fic ance . Fourier T ran s form Method :
T he t heory of thi s method is beyond the
scope o f this outline an d the reader is referred to the speciali zed literature ( K l u g and Alexander , 1 97 4 ; C o he n and S c h w ar t z , 1 97 7 ) . T he experimen tal determinations are s i milar to those outlined above , but t he p ure di ffrac tion p ro file is c alculate d from t he Fourier synthesis of the sample p rofile ,
h( £) ,
si ze ,
an d a re ference profile obtained wi t h a reference sample of in finite g(
£) ,
whe re
£
i s t he an gular chan ge throu gh the p e ak .
T he Fourier tran s form coefficients o f the true diffraction profile s are calculated from the real and imagin ary Fourier serie s c alculate d from t he inten sitie s of sample an d re fe rence profile s meas ured at various an gular in tervals o f the broadened peaks .
Computation procedures are gi ven
Gane son et al .
( 1 97 8) and Sashital et al . ( 1 9 7 7 ) . T he crys t allite size c an be c alculated u si n g t he breadth of the recon structed p rofile an d S c herrer' s form ula or directly u sin g t he Fourier coefficie n t s . W arren and A verbac h ( 1 950) relate d t he crystallit e si ze to t he fi r s t derivative of the Fourier coefficie n t s wit h respect to the Fourier harmonic . Furthermore , a plot o f the second derivative o f the Fourier coefficie nt is p roportional to t he dis tribution of c ry s t allite si ze s . Fi gure 2 sh ow s re sults of Sas hit al et al . ( 1 97 7 ) o b t ained u sin g the Fourier tran s form metho d , includin g strain and imperfection correction s .
T able
2
s u m m arizes t he average crystallite si ze
for the various c rystal planes alon g wit h dispersion result s obtained from hydro gen chemisorption technique s .
I t c an be seen t hat t he Fourier
tran s form c alculation s agree w ell wit h the chemi sorption res ult s .
T ra n sm i s si o n E l e ct ro n M i c rosco p y
T h e most com mon use o f an elect ron micro scopy ( EM ) in cat aly si s is to meas ure t he dist rib ution of cryst allite si zes as well as the m o rp holo gy of supported or un supported active m at e ri als . tron microscop e s , w hich combine
EM
Howeve r , new an alytical elec
w it h spectroscopic analysis , also re
veal in form ation about the chemical nat ure of the c rystallit e s .
To under
s t and t he pos sibilitie s of EM beyond t h at of takin g pic t ure s , it is
nec e s s ary to have a basic understandin g of E M .
A detailed description o f
t h e s ubj ect i s presented in various textbooks ( H eidenreich ,
1 97 2 ) and review article s ( B eer , 1 9 80 ;
S chmid t
et al . ,
1 96 4 ; H awkes ,
198 2 ) .
1 62
De lgass and
�
3
' ' ' 'I '
�
( u,,
'
: I
Q 2
I
!' I
..
N c
�
' ' '
11,
' '
.., .... 4.c I N ..,
0
Wolf
0
10
.d
tJ
)>
b,
//
20
'!�..
"·
30
b- • • ..,
40
50
(a) 8
2
-0 ..
;,
c
4. "'
'0
0
b
0
'•
40
50
� I
N
,o
A ...._� ..._ __JL._____L__
0
10
20
0
30
40
L = no 3 , A
(c)
F I G U R E 2 Particle si ze distribution for a catalyst containin g 40% Pt on Si0 2 prepared by impre gnation of chloroplatinic acid : ( a ) < 1 1 1> direction ; ( b ) < 1 00> direction ; ( c ) < 3 1 1> direction .
An electron microscope differs from an optic al microscope in t hat the electron beam rep laces t he li ght be am an d electrom agnetic lenses replace T he use of an electron beam imp rove s t he resolution the op tical len se s . of the microscope , which is p roportional to the wavele n gt h >.. of the inci dent beam , in accord with ( Heidenreich , 1 964 ) ( 12)
w here x is the resolution ( i . e . , the m1mmum si ze t hat can be distin guished in t he microscope ) and c8 is the sp heric al aberration of the incident beam . R elation ( 1 2) illu strate s the advantage of usin g an electron beam ( A 0 . 5 to 10 A ) in ste ad of li ght ( A = 40 0 0 to 7000 A ) . =
1 63
C ataly tic S urfaces TABLE 2
C a talyst
P ar ticle Si z e s
a
7 . 1- SiO - PtC l - 8 2 ( 1)
< L > hkl
1 1 1 (A)
a n d D i s p ers io n
< L > 1 00 (A)
uo
311
Dx
Dh
10 12
7. 1
21. 5
( %)
13 1
82
111
75
( 1)
49 48
43 43
45
47
25
( 2)
( 1)
43 40
39 39
42 39
31 39
27 29
22
20
44
( 2)
2 1 . 5- S i0 2 - I onX - L 2 7 - Sio 2 - IonX - s
( 2) 40- Si0 2 - Pt C l - S
( 1)
30
26
( 2)
25
23
108
45
81
47
( %)
7. 1
25
27 . 3
39. 8
47
a
7 . 1 , 2 1 . 5 , 2 7 , an d 40 indicate percent of P t loadin g o n Si0 2 . PtCl and IonX in dicate preparation vi a c hloroplatinic acid im p re gn ation and ion e x c han ge , S and L refer to Si0 2 w i t h sm al l ( 1 2 0 t o 1 40 ) and lar ge ( 7 0 to 8 0 ) mesh si ze . D x i s t h e di sp er s ion obtained from x - r ay line broadenin g , and Dh is t he value obtained from hyd ro ge n c hem i sor p tio n .
Two fundament al models of operation are u se d in EM : transmi ssion electron microscopy ( TEM ) and sc ann i n g electron microscopy ( S EM ) . A t hird , hybri d mode , scannin g transmission ele c t ron m i c ro s co p y ( S TEM ) , is also used . Fi gure 3 d e p ic t s the schem atic s of the various modes of oper a t ion t o ge t h e r with the fundament al component s involved in each case . In t r an sm i s sio n electron microscopy ( TEM ) the elec t ro n beam , ge n era te d by a heated filament ( electron gun ) , p as s e s t hrough two electrodes and a con denser len s . P arallel r ay s thus cre ate d impin ge on t he spe c i m e n , where they are scattere d as a re sult o f t he spatially v ari ab le refractive inde x . Scattered ray s from t he same point in t he s p ecim en are b ro u gh t to t he T he overall effect same p oi n t in t he i m a ge formed by t he o b j ec ti ve lens . is eq uivalent to transmisssion o f primary electrons t h ro u gh t he sample . Electrons are also diffracted ( as in the case of x ray s ) , thus producing rays which are s li ghtly off the angle of r ay s formed by t r an s mi t t ed elec trons . Thus , by selectin g the p r op e r ap e rt ur e , an i m a ge of the trans mitted ( b ri gh t field ) or di ffr act e d electrons ( d ark field ) is obtained after the electrons pas s through intermediate a n d p roj e ct o r lenses and i mpin ge on a fluorescent screen or a p ho to graphi c plate . A typi cal con ventional electron m icr o s cop e operates at 1 0 0 kV , a vacuum of to 1 0- 6 torr , and is c ap ab le of 4 . 5-A point - t o - p oin t resolution and man gific a tion of 3 0 0 , 00 0 . D edicated hi gh - re soluti on electron microscopes are c ap able of re soluti on of 2 . 5 A with magnifications of 800 , 0 00 to 1 , 0 0 0 , 000 . An im p ort an t aspect of TEM is that a three - dimensional s ample yi elds a two- dimensional TEM i m a ge w hi ch m i gh t be difficult to interpret . Further more , the contrast between the s u pp o rt an d t he active component mi gh t
1o- 5
164
De l gass and Wo lf E l ECT RON S O U II C E
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S P E CIMEN
� AMPLIFIER
OPTICAL
- - - - - )>
M I C II O S C O PE
A P E II T U II E
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SCANNING COILS
fV'V"" I N T E II II E D I ATE LENS �
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SPE C I M E N
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---- �
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E N E R G Y DI SPEIISI VE SPECTROMETER
(0 )
TEM
( b)
SEM
(c) EPMA
Schematic s of ( a) transmis sion electron microscope ( TEM ) : ( b ) sc annin g electron micro scope ( S EM ) , an d ( c ) electron probe microanaly zer ( EPMA ) .
F I GURE 3
not be sufficient to distin guish the m etal crystallite s . In this case , dark field images are useful since electron s diffract p referenti ally from the cryst allit e s , thus producin g b ri ghter spots from them com p ared to the weak diffraction from the nearly amorp hous s upport .
S up ported Catalysts Fi gure 4 shows bright- and dark - field i m a ge s of a supported 5% Pt / Si0 2 c at alyst . T he s ample is prep ared by mortar grindin g the catalyst particles ( 2 0 0 m e s h ) to ultrafine pow der , w hich is then dispersed in a liquid ( w ater , alcohol ) . A droplet of the solution is then deposited on an EM grid . The liquid evaporates , leavi n g on the grid microscopic p articles which are thin enough for electron tran s mission . T he micro grap hs shown in Fi g . 4a were obtained wit h a 100 JEOL S TEM operating at a m a gni fication of 4 0 0 , 00 0 . I n this c ase , the contrast i s relatively low because o f the presence of the support . Counti n g and si zin g the crystallites in both t he bri ght - an d dark - field im a ge s is o ften diffic ult , and m an y micro grap h s are requi red to obt ain a s t ati sti cally meanin gful si ze distribution . A m ajor limit ation here can be the p re sence of s m all crystallites that cannot be seen under the Furthermore , in some cases the diffraction p attern of con ditions used . the metal c annot be distin guishe d from that of the s up port ( S chmidt et al . ,
1 65
Cataly tic S urfaces
(a) 25 0 A : FIGURE 4 TEM photograp hs , all at magnification 400 , 00 0 ; 1 e m ( a) 5% Pt /Si0 2 ; ( b ) model sample , Pt sputtered on Si02 film ; and ( c ) model sample , Pt impregnated on Si0 2 film with chloroplatinic aci d . All catalysts were reduced in H 2 at 300° C . =
1 66
Delgass and Wolf
(b) FIGURE 4
( Continued)
Catalytic Surfaces
1 67
(c) FIGURE 4
( Continued)
1 982) . It is always recommended that one check the electron diffraction pattern to determine the relative role of the support and active metal in dark- field studies . C har gin g of the sample by the electron beam is a com Chargin g mon problem with catalysts in which the support is an insulator . can be reduced by coatin g the sample with a carbon or gold film . Higher metal loadings ( 5 to 1 0%) and higher beam voltages result in better con Sam trasts ; however , sample damage can be caused by the electron beam . ple preparation procedures are given by B aker et al . ( 1 979) Glass! et al . ( 1 98 0 ) report artifacts introduced due to sample preparation . The particle size distribution is obtained by the method described by Underwood ( 1 97 0 ) o r b y means of a n automatic sizing and counting device . Model Samples Schmidt and coworkers ( Schmidt et al . , 1 982) , among others , have dis cussed the advantages of usin g model samples in which the catalyst is
1 68
De lgass and Wo lf
d ep osi t e d on a planar nonporous substrate of a morp ho u s silic a . A thin flake of sili ca is p rep ar ed by vacuum de po sition of silica fo llo we d by vac u um deposition of th e met als ( e . g . , Ni , Rh ) onto the sub str ate . This t e c hni q u e avoi d s p ar t icle over l ap an d p ro duce s hi gh co ntr a s t between the met al p ar tic le s and the s u pp o rt Figure 4b show s TEM r e s u l t s obt ained by s p ut terin g Pt onto a model silicon grid and Fi g . 4c sho w s similar re s ult s obtained by i m p r e gn ati n g t he gri d with H 2P t C L s an d then re d u cin g it in H2 at 4 0 0° C . T he di s adv an ta ge of t he technique is that the to t al active area is lo w for re action studie s . F u r t her m or e the morp holo gy of t he p ar ticle s de p o si t e d on the model , nonporous support , m ay be di ffe rent t han w hen t he metal i s de po site d on p o rou s si lic a . A l t h o u gh TEM has been used extensively to stu dy cataly sts , in conven t io n al op e rati on it views s am pl e s in a vacuum environment , pre or p os t re action . A te c h niq ue that allow s o n e to use TEM in a reactive environment is controlled at mosp he re transmission electron microscopy ( C ATEM ) , De t ails of the CATEM te ch nique have been do c um e n ted in a review arti cle by R . T , K . B ake r ( 1 97 9) , w h o has intro d uced t he te c h niq ue to c atalytic re s ear c h ers In CAT EM , t h e sp eci me n is m ounted in a gas re action cell w hich is fi xe d to the translation al m an ip ula tor of the sample . With this arran gement it is p o ssi b le to operate at gas pr e s s u re s of t he order of s e ve ral torr within t he cell and at t e m p er at ur e s up to 1 300° C . T hi s con trol of sample environment re qui r es some sacri fice of resolution . C h an ge s occ u rrin g durin g reac ti on are recorde d b y a TV video sy st em B aker ' s e xce l len t paper ( 1 97 9) i ll u s t r at e s the use of CATEM i n s t u dyi n g gas - soli d systems w here the reaction prod uce s c arbo n de p o s its o r d uri n g gasification of t he deposit s , which induces motion of small metal p arti c les and c han ges in t he structure of the spe ci me n . E xt r aor di nary movies have been pro duced s how in g the growth of carbon filame nts as well as the mo ve ments of met al p ar tic le s d urin g c arb on gas ifi c atio n S uch i n fo r m ation has reve aled the com p le xit y of t he dy n am ic s of gas - solid i n t e r ac tio ns ·.
,
.
.
.
,
COM PO S I T I O N A N D S T R U C T U R E O F S U R FA C E S A N D S O LI DS
Since the 1 960s , several techniques h av e been de ve l ope d to c har ac t eri ze A gr e at deal o f effort has b e en spent on the measurement of surfaces . basic prop e r tie s o f model s u r face s ( such as single crystals ) to establish t he theory an d va li di t y of each surface an al y si s met hod . T h es e st udies have le d , in some cases , to int e rp r e t ation of c at al yti c act i vi ty in t er m s of fun dament al p roper t i e s and h ave demonst r ate d the potential of these tech T he di s c u ssi on t hat follow s is niques in s t u die s of t e c h n ol o gi c al interest . p r es en te d in t he context of t he app lic ation of t he se techniq ues in the i nt e r p r e t atio n of s ur face p rocesses relate d to catalysis . T heoretical a s pe c t s are treated only on the m o st b asic term s needed for i n t e r p re tat i on of dat a . T he reader i s re fer r e d t o t he s p e ciali z e d literature for fu rt he r de t ail conc e r nin g theory . an d for det ailed discussion of e q ui p me n t and opera t io n Most of the te c hni q u e s no w com m e r ci ally available ar e b ased on the e x cit ation o f t he sample by a n in co mi n g beam o f p article s or electromagnetic energy . Followi n g such e xcit atio n are d e - excitation p rocesses , w hic h re sult .
in t he emission of e le c t ro ns or p hotons of di s c ret e energy w hich can t hen be me asu red by a suit able ene r gy analy zer . F i gu re 5 s ho w s the s c he mat ic s
1 69
Ca taly t ic Surfaces Visible or U V Source
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oy:
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X- r o y P ho t o elect ron ( X PS)
ATOM
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K 011 R a d 1 o t i o n
'ZVTfl/ < a l
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V22?JZ ( by-
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FIGURE 19
Splitting of a spin-1/2 n ucleus in a ma gnetic field .
100 gauss ,___,
"o''o
........ .. _ ..�. ..__-L..---J'--'---'--..__ ..__.P7 ...._ o 17o
�..--
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FIGURE 20 First derivative ESR spectrum of 0 2 on t he surface of hi gh surface-area MgO . T he field increases from left to ri ght and the gain has been reduced five fold for t he central line ; a small portion of the high field spectrum is overmodulated to show t he outermost lines . For clarity no attempt has been made in this diagram to insert levels for t he lines aroun d g1 . ( From T ench and H olroyd , 1968.)
201
Catalytic Sur'faces
help ful in d efinin g the en vir onme nt of an electr on spin. As shown in Fig . 20, the i s ot ope 17Q, w ith a nucl e ar spin of 5/2 shows 6 line spli ttin g s for 17Ql6o b u t 11 line s pli t t in gs for 17o2 ( Ten ch and Holroyd, 1968). Note that the s p e c tr um is di s p layed as the derivative of the energy ab sorption curve, as is standard in ESR spectroscopy. NMR has s en si tivi ty to chemical b ondin g through shieldin g effects. The more free electr ons around the nucleus of i nt erest are to circ u late in response t o t he app li ed m agnetic field, the more they decrease th e st ren gth of the field felt by the nuclear spin. The ran ge of chemical shifts in duced by this shie ldi ng is ge n erally large with respect to NMR line widths in liquids , giving t he method high chemical s e nsitivity. Spin -spin cou pl in gs betw ee n n onequivalent nuclei als o occur, giving NMR furt her c ap abil i t y for de fi nin g the chemic al en vir on ment of a give n type of nuc leus . In li qui ds , many interactions are a vera ged to zero because of the rapid m o tion of the molecules. In solids, however, rel axation times are long and direct - dipo le co up ling of the s pin s and anis ot rop y of the chemical shift t e n s or both tend to broaden lines to such an extent that meanin gful chemical s hi ft information An alysi s of t hese two in cannot be obtained from a standard s pect rum . ter actions s hows that if t he sample is s p in nin g on an axis 54°44' to the ma gneti c field direction, t he static components of the dipole-dipole inter action and the c he mical shift an i sot r op y both b ec o me zero (Andrew, 1981).
S i d eba n d s , whose p osi t i ons depend on rotational frequen c y , still appear but do not obscure t he s pectrum . T his tec h ni q ue , called magic angle
spinning, drama t i ca lly improves the resolution of NMR of solids (Andrew,
1981). Further impr ovemen t s can be made wit h spin decouplin g , c ro ss polarization (Yannoni, 1982), and m u lti p le p ulse te c h ni ques ( Tay lo r et al., 1980; Ryan et al., 1980). In the p ast, NMR ha s been a p o we rful tool for c h e mi c al st ru ct u re a n alys is . The potent ial of the new NMR method s is to bring s u c h detailed s tr uc t ural information to the study of catalytic solids. NMR App licatio ns
Magic angle s pi n nin g NMR has provided a vari ety of new o p p o rtunities for stu d yi n g the d et ai le d structure of zeolites. As shown in Fig. 21 , 29Si NMR cle a rly resolves Si with zero to four Al n eigh b ors bo n ded th r ou gh oxyge n bri d ges . In this work by R amd as et al . ( 1981 ) , the distribution of AI neig hb or s a round Si wa s st u di ed as a function of Si/Al r at i o for synthetic faujasite s . They found that Loe w e nstei n ' s rule forbi d din g AI atoms from occupying nei g h bori ng tetrahed ral sites was obeyed . As t he Si/Al ratio approached 1, the NMR spectrum became a single line cor res p ondi n g four AI nei g h bo rs around Si. By analyzin g t he relative in tensit ie s of lines as a fun c tion Si/Al ratio , the authors const r u c te d an or d erin g scheme for Al pl aceme n t in the fa uj a si te lattice . The on se t o f pa r a p osi tionin g for Al in the hexagonal ri ngs facing the superc ages was found to begin at Si/Al r at io of a b out 1 . 4. The simulation in Fi gu re 21 is based on thi s structural m od el. It is inte r esting to note that Loewenstien's rule appears to be broken for zeolite A ( B ursi ll et al., 1980). T ho mas and c ow orkers have also used solid - st ate NMR to show the s i mi la ri ty of ZSM-5 and Silicalite (Fyfe et al., 1982). Nagy et al. ( 1981) have observed that the chemical shifts for ZSM- 5 and sim ila r zeolites are more ne gative t han those o f the fauj as i tes and have ascribed that dif f e re n ce to the presence of the fi ve - membered rings. In another study,
a
Delgass
202
and
Wolf
4
3
4
3
Si/AI
=
1.35
0 90
FIGURE 21
li[p.p.m.l
100
110
Observed ( at 79.6 MHz) ( upper) and simulated (lower) high resolution 29si NMR spectra with magic an gle spinnin g for a Si/Al ratio of 1. 35 in synthetic fauj asite . ( From Ramdas et al . , 1 98 1 . ) Nagy et al. (1982), u sed cross polari zation ( couplin g of t he proton and 29Si resonances ) to identify silanol groups in the spectrum o f ZSM- 5. Alt hou gh magic an gle spinnin g is certainly important t o applications o f solid- state NMR to catalysts , it is not always necessary. Stichter ( 1981 ) has reporte d a very interestin g set of 195pt NMR experiment s on Pt/Al203 samples supplied by Sinfelt . T he data were obtained by t he spin echo technique. T he lines observe d were very broad , about 4.5 kG, the ran ge of che mical shift between diamagnetic Pt compounds and Pt metal . T he large shift o f metallic Pt , called the Knight shift, arises from the polari zation o f t he spins o f t he conduction electrons . Analysis of the peak shape as a function of Pt dispersion ( fraction exposed) shows t hat the amount of diam agnetic Pt is directly related to dispersion . T he spectra were interpreted to indicate that adsorption of gases on the sur face ti e s up the otherwise free electrons of. t he metal and e ffectively demetalli zes the s urface atoms. Direct observation of such fundamentally important p henomena can h ave a great imp ac t on our understandin g of catalysis . Fourier trans form 13c-NMR o f species adsorbed on catalyst surfaces has also been effective . Nagy et al . (1982) have been able to follow the isomerization of 1- butene to 2- butenes on a tin- antimony oxide catalyst. Gay ( 1 974) ob served a variety of molecules adsorbed on silic a and ex amined their mobility . Cirillo et al. ( 1980) h a ve used proton NMR to
203
Catalytic S urfaces identify three types of low- te mperat ure molybdenum/alumina catalysts.
H2
adsorp t ion
on alumina and
DISTRIBUTION OF CATALYTIC MATERIALS WITHIN CATALYST PELLETS:
ELECTRON MICROPROBE, SCANNING ELECTRON
MICROSCOPY,
AND ANALYTICAL MICROSCOPY
n
tant y i and no n main tools
of materials ins t poisoning due to metal such analysis are the l ng
A impor variable in the performance a catalyst is the distribution of catal t c nc atalytic ide the pellets. The latter is significa t in the case of ca talys deposit o . The now available to provide e e ctr n probe microanalyzer (EPMA), scan ni electron microscope ( SEM) , and scanning transmission electron microscope (STEM). The principle of oper tio n in all three cases is the but the EPMA is designed to ve best resolution for chemical a alys s , whereas the electron microscopes are designed to high and good p a and depth resolution, with chemi cal analysis b i g added capability. The SEM p r od es two-dimensional e ect on backscattered from the material instead of from elec trons transmitted through the material, as is the case in T EM . A STEM w th sca ni g capabilities and also be used for chemical yno p i that follows will briefly describe the principle of emission for chemical i these instruments and then present description of scanning electron microscopy. A brief co pari the EPMA SEM presented. Fi ally, applications t cat i
i n
o
a
same, n i give magnification e n an images from l r s
is a TEM i n n analysis . The s ss x-ray used a son of and alys s are described.
gi
s ati l
uc
can
analysis n
m
is also
n
o
Chemical mi c roa al y is is carried out by an y is of the energy x rays emitted when a sample is excited by electron i pact or x rays. The phenomenon has already been referred to p re vio s sections in connec tion with sources for x-ray diffraction and photoelectron e s io . x rays are used as the xcitati n so rce , the ec niq e is .r-ray fl uores ce nce analysis. S uch instruments are pr ril y for chemical l ys s s mples because x ys are if icu lt to and there fore the spatial e l tio as good w he electron impact is the excitation source. Electron beams can be focused to diameters fro 25 1000 A. F rtherm re , since electron beams are the primary excitation in e lectron icros op y, x-ray analysis constitutes a natural additional capability for such i stru e ts . Figure 5b shows that w he e lectr is remo e inner shell of an be filled by an electron an outer shell. The energy be emitted as an x-ray phot . Deexcitation by A ger electron emission can also occur. Since levels are quantized, it fo l ow that the energy of the emitted x-ray pho t n is discrete and is characteristic of the e le me t which it e it ed . Consequently, n w e ge the emitted x-ray energy be used for chemical characterization of the ele en . If in the K the L level , photon is designated as Ka (Ka Ka2 exist
Principles of X -ray Emission or X -ray Fluorescence
n
s
al s m in
e
ana
o
u
i of bulk a r so u n is not
ra
as ranging
m
to
u
source
of
u
mis n t h u called used ima d f focus as n
o m
When
used
c
n on
m n n an v d from an atom , creating a vacancy , it may from released by the transition may on u the energy l s o also a quantity n from was m t k o l d of can m t the vacancy occurred level and the transition occurs from the and due 1 to differences in spin ). Such emission gives rise to characteristic lines
Delgass and Wo lf
204 s up erimp os ed on a co ntinuo us spectrum.
The characteristic lines produce quasi-monochromatic x rays which are us ed as the source in XRD . W hen x ra ys are used as an ex citation source (x-ray flu or es cence ) , no con ti nuou s spectrum is observed, and the background is very low, which is esp ecia l ly suited for energy ana lysis . The additional pro cess es res ultin g from e le ctro n impact are shown schematically in Figure 22. S eco n dary and transmitted electrons are used to produce im ages in SEM and TEM, respectively. Auger electr ons are used to obtain s urfa ce chemical information. Energy analysis of the char acteristic x rays also provides such information. Also shown in Fig. 22 is the depth of escape of the v ario us deexcitation processes. It can be seen
that the escape de pth of x rays is on the or der of micrometers ; there fo re , they do not con t ain surface information (as in the case of Auger elect ro n s) . X -ray em iss ion is genera ted in a v o lume of about 1 \1m3 , w hich makes x-ray analysis a m ic ro analytical tool . (a) w ave le ngt h dispersive spectrometry
Two methods are us ed to analyze the energy of the emitted x rays: (WDS), and (b) energy dispersive
spectrome t ry (EDS). In the former met hod , the energy of the x ray is obtained from knowing its wavelength (E = hv hc/A). A is determined usi ng a crystal of known crys tal spa ci n g which diffracts (disperses) the inc oming x rays, allowing only one wavelength element to reach the de The pri nci ple is the s ame as d es cri bed in XRD, a nd Bragg's law tector. is used to determine A . Energy dispersive analysis (EDS) i s a cco m plishe d by u si ng a semiconductor crystal that absorbs x rays, thereby exciting the crys t al ' s electrons into the c ond ucti on band, producin g ari electr ic pulse proportional to the energy of the x ra ys . These pulses are then counted and sorted in a multichannel a nalyz er , and a displa y of in te nsity of the signal versus its energy gives the desired spectrum (see Fig. 23a). The advantages and di sa dva nta ge s of these two· types of x-ray energy analyzers are s umm ari ze d in Ta b le 4. =
Incident Beom BSE
1!t
CXR
Surface
AUQer Electrons (AE)
lf""�+-B -+--�1"-- ock Scollered Electrons (8SE) lmicron
Characteristic X- roy (CXR) X- roy Continuum (XRC)
FIGURE 22
S c he m at ic of various signals originated by electron impact,
showing the co rrespo ndin g volume of p rimary e xc it a tio n .
205
Catalytic Surfaces
CoKa
Si Ka
> t-
ii) z I.LI t z
ENERGY
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jj;
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e
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0.04
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•-
0.16
(.)
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FRACTIONAL RADIUS
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X-ray spectrum
FIGURE 23 (a) of a sample containing Si, K , C o, and Ti obtained by EDS. (b) Vanadium concentration profiles on a 1/16" extrudate catalyst during hydrotreating of a heavy oil residuum. Effect of pore di ameter. (From Tam m et al., 1979.)
TABLE 4
C omparison of EDS and WDS
C omp act : low cost Rapid (qualitative analysis) Simultaneous multielement analysis of the fu ll x -ray spectrum (Na upward) Display of the entire spectrum in digital format
A dvantages of energy dispersive spectroscopy
Low sensitivity to geometric effects High collection efficiency
Lack of higher-order lines which are generated in crystal diffraction Digitally produced outputs for element line scans and distribution maps Advantages of wavelength-dispersive spectroscopy Higher inherent element separation (resolution ) High count rate on individual elements Analysis can be highly quantitative B etter peak-to-background ratios An alysis of wide range of elements (Be to U)
a
Higher sensitivity Operates at room temperature
206
Delgass and Wolf
Instrumentation
A brief discussion of the major components of the various instruments used in electron probe microanalysis and SEM is given below; several books (Postek et al., 1980; Goldstein and Y ako w it z, 1975; Thornton,
1974)
1968) and
review articles (Sargent and Embury, 1976; Johari and Samudra,
on
SEM, and EPMA (Heinrich, 1980; Purdy and Anderson, 1976) are available for a more detailed description. Fi gure 3 shows the schematics of a TEM, an electron probe microanalyzer (EPMA), and a SEM. These instruments are equipped with a system to produce and focus an electron beam and with a system for detection and analysis of x rays.
The EPMA is designed primarily for x-ray analysis, with
an electron-beam intensi ty which optimizes the x -ray yield and a low-magnifi cation optical microscope for viewing the sample.
100, O O Ox
The SEM, on the other
A.
hand, is designed to obtain images which, under ideal conditions, can be magnified in the 20x to Because
of
tail in the
range with resolutions of 70 to 100
the greater ve r sat ility
followin g
of
the SEM, it is described in more de
paragraphs.
The SEM differs from the TEM in that ima ges are produced from second ary electrons generated by the
electron
beam.
Thus the sample need not
be transparent to electron transmission as in TEM. a
volume
of primary excitation (see Fig.
Electron impact creates
22) from which elastically scat
tered ( backscattered) a nd inelastically scattered electrons or secondary electrons are emitted.
Secondary electrons have low energy (less than
50 eV) , and therefore only those electrons near the surface can be emitted and detected.
Secondary and backscattered electrons are emitted outward
in all directions.
An electron detector system is used to collect both
types of emitted electrons and produce a signal proportional to their in The detection of secondary electrons depends on the sample
tensity.
topography.
Electrons emitted from the specimen surfaces facing the de
tector will show a bright face, whereas those emitted from surfaces facing away from the detector will not be detected, creatin g a shadowin g effect. Furthermore, as the name indicates, an SEM is equipped with deflecting coils which permit scanning the electron beam over the sample in a raster mode.
The raster pattern of the primary electron beam is synchronized
with the scannin g pattern of a cathode ray tube (CRT) used to display the detector signal so that a
produced .
t wo-dim ensional
Images are produced in the CRT
records can be
obtained
image of the scanned area is
or
a TV, and permanent
directly from the CRT usin g an instant camera or
by videotape recording from a TV.
Of the two types of x-ray energy analyzers, WDS is the preferred mode of analysis from the standpoint
ly
of
x -ray spectroscopy and is used exclusive
in x -ray spectrometers as well as on EPMA instruments.
WDS analyzes
one wavelength element at a time and requires mechanical components to rotate the crystal as
well as
electrical components.
Crystal alignment is
quite important, and WDS is difficult to use with rough surfaces. analyzers are used in most SEMs and in some electron probes.
of and
EDS
EDS pro
vides a faster and easier survey analysis
the elements but at lower
energy resolution and lower sensitivity,
it does not cover the entire
spectrum, especially of light elements such as carbon.
Clearly, for
versatility one would like to have an SEM capable of usin g hi gh beam intensities such as in EPMA an alyzers.
and
equipped with both WDS
and
EDS
For hi gh-resolution microscopy work or x-ray spectroscopy
work, separate instruments are preferred.
Figure 23a shows the x-ray
207
Cat alytic S urfaces
spectra of one spot on the sample. However, the be a m can be scanned over the sample with the analyzer fixed on a certain energy so that only x rays cor respondin g to a sp eci fi c element are disp l ayed This mode o f .
oper ation pr o duces an x-ray m ap of the element in ques t io n over a line Another w ay of o b t ai ni n g elemental ( line scan) or area under an a lysis distribution is to move t he sample under the fixed electron beam. This technique is particularly useful when analyzin g catalyst pellets because it per mits one to scan t he entire width of the pellet. For electron m i c ro analy si s the sample m ust have a flat s ur fa c e . For powders, the sample is first embedded onto a p last ic medium, w hich , afte r drying , is ground to reveal the cross section of its i nter io r . Con duct ive coatin gs ( C, A u) are evaporated onto the surface of nonconductive materials (Purdy and Anderson, 1976) . For SEM, the sample is mounted on a small stub coated with adh e siv e wh i ch ho l ds th e par t ic les as received .
,
or after fracturin g to re v eal t he interior.
Speci fic applications of SEM and E PMA to ca t al ysis are di scussed by
Applicatio ns
Sargent and E m b ury ( 1976) and Purdy and Anderson ( 1976) . SEM is used in general for ph ysica l c haracterization of c at al yst particles, s uch as particle size, sh a p e s urface s tr ucture and poro s it y makeup of aggregates , a nd dispe rsio n of one phase into ano ther The practical m a gnif i c ation and ,
,
.
resolution of the SEM is r el a t iv ely low compared w ith the TEM and STEM, which are pref erentially used for e x am i nin g crys tallit es in the size range 1 to 10 mm. X-ray em issio n cap abilities of the electron microprobe or SEM instru ments are used to obtain bulk chemical information about the materials present in th e s ample . The most import ant ap pli cat ion is to me asure the distribution .of catalytic materials or i mpuriti es over the cross s ectio n of c at al yst pellets. A cl assi ca l example is t he effect of catalyst p rep a r a tion in the dist ri but ion of the active catalyst inside pellets . The metal profiles insi d e the pellets can be followed as a func ti on of preparatio n c onditions Dep en di n g on the m etho d used t h e c at alyst can be depo sit ed u ni for ml y inside the pellet , or it can concentrate toward the external face of the p el let Extension of this application will increase o ur understanding of catalyst p repa rat ion. Another classical ap p li catio n is the st udy of metal d eposition oc c u rri ng d ur in g dehydrodes ulfurization of he a vy residuum (Tamm et al., 1979). Fi g u r e 23b shows the V cont ent of a catalyst as a fu nction of po sitio n in side th e p el le t , a s o bt ain ed us in g an EPMA . The p o is on i n g of catalyst s due to impurities is an area of study in which EPMA and SEM are pa rti c ular ly useful tools . .
,
.
ADSORBED SPECIES ON SURFACES Infrared Spectroscopy
Unlike the surface a n a lysis t e c hniq ues based on electron spectroscopy, i n fr ar e d (IR) s p ec troscop y is a well se asone d tool for catalytic research, as witnessed by the v a rio u s books (Little, 1976; Hair, 1967; Clark and -
Hester
P lis k in
,
,
1974-1982) and review articles (Del gass et al . ,
1958) published on the s ubj e ct
.
Eichens
IR spectroscopy was firs t 1979;
and
Delgass and Wolf
208
developed to study t he chemical structure of bulk compounds, but it was soon re alize d that it can also be used to gain s tru ctura l in fo rmation abo ut s p eci es adsorbed on solid surfaces. The books by Hair ( 1967) and Little ( 1 976) contain a good introductory description of IR to get he r with m any ap The review article by Eichens and Pliskin ( 1958) has withstood plications. the test of time and remains one of t he most quoted re f erenc esThe more recent review arti cle s (Delgass et al., 1 979; Basila, 1 96 8; B lyhold er , 1968; Pritchard and C atte rick , 1976) e mp h asi ze newer developments. The descrip tion give n here cov ers only the most fundamental aspect s of IR spectroscopy and its maj or applicatio n s to cataly ti c research . IR spectrosc opy gives qualitative information abo u t the way adsorbed molecules are bonded to sur fa ces , as well as structural information on solids. It can also be used to measure the amount of materials some case s it can be used to stud y the ra te at which certain surface proce sses occur.
adsorbed, and in
Principles of Vibrational Spectroscopy When an atom or molecule is p laced in an elec tro ma gn et ic field , there is a tr ans f e r of ene rgy between the field and the quantized energy leve ls of the mo l ecu le s . When energy is adsorbed , me asurement of the transmitted energy versus the frequency of the radiation produces a spectrum uniq ue to the excited molecule. If the frequency of th e incoming radiation is in the IR range , the spectrum is due to t ran siti ons in vi b ration al and rotation al energy
levels. In each electronic energy level there is a series of vibrational transitions, and for each vibrational level there is a series of rotational Transition in electronic ener gy levels occurs when t he excita transitions. tions have fre quenci e s in the UV or visible rangeSelection rules govern t he perm i ssibl e transitions amo ng the various energy levels.
Diatomic Molecule: Vibrational spectra arise fro m the motion of one atom relative to another. While a detailed group the or y has been developed to describe the phenomena, a simple explanation in whic h a diatomic mo lecule is mod ele d as an harmonic o s c illator is often adequate to describe the first ord er processes involved. The energy and f re quenc y of an ele ct r omagne ti c wave are related by E = hv, where his Planck's c o nst an t . Since AV c, where c is the s peed =
of li ght , it has be co m e a common p r acti ce to r ep ort spectra in terms of frequency i ns te ad of AThe unit most commonly used is the wa v e n umber 104/).. (micrometers ) , which has units of cm-1. The IR spectrum is divided in t o near IR, 10,000 to 5, 000 cm- 1 (A: 1 to 2 ]..lm); mid IR 5000 to 200 cm- 1 (A: 2 to 50 11m), and far IR, 200 to 10 cm-1 (A: 50 to 1000 IJID). Solving the classical mec h anical eq ua tions descri bin g the oscillations of two masses, m 1 • m2 , attached to a perfect spring which obe ys Hooke's law, one can calc ul ate the change in potential en ergy associated with the oscilla t io n s as =
u
( 21)
where k is the force c onst an t that measures the stiffness of the spring and r is the displacement from the center of mass. The motion of this system is sinusoidal and the fre quency is given by v = ( 21T)-lv'k/ll, where 11 = m m /Cm + m ) is the reduced massThe simple result of the harmonic 2 1 2 1
209
Catalytic Surfaces
oscillator has immediate practical applications in the interpretation of spectra. Knowing v, one can calculate the force constant k; conversely, changes in v upon adsorption reflect weakening (lower v) or strengthening (higher v) of the bond. Another application of the result obtained for the simple diatomic oscil lator is in connection with changes in frequency due to isotopic substitution. If the force constant remains the same, then upon isotopic substitution of species with reduced mass � 1 and a frequency v1 by an isotope of reduced mass �2, a frequency change predicted by
( 22)
should be observed. Failure to observe a frequency change indicates that the assumed species or group is not involved in the vibration being ob served, and another structure must be postulated [see Delgass et al. (1979) The potential energy and Hair ( 1967) for illustrations of this application]. diagram described by Eq. ( 2 1) is similar to the lower region of the energy diagram of a real diatomic molecule. For this reason, the quantized form of the harmonic oscillator analogy is adequate for a simple description of vibrational spectroscopy. To account for quantum effects, the potential energy function of the harmonic oscillator is substituted into Schrodinger's wave equation. The solution of this equation gives the energy of the quantized vibration as ( 23)
E
where v is the vibrational quantum number. Certain selection rules are used to ascertain the probability of transitions between energy levels. For ±1 are permissible. harmonic oscillators, only transitions between 11v Since at room temperature most molecules are in the ground state, vibration al transitions between ground state and the first energy level are usually observed. Another important selection rule requires that a change in dipole moment with vibration must exist in order to couple the vibration to the electromagnetic radiation. In the absence of such a dipole moment change, as in the case of linear diatomic molecules containing the same atom (i.e. , N 2> , no vibrational energy transitions are observed. A corollary of the foregoing selection rule is that IR inactive molecules may give rise to dipoles when they are adsorbed on a surface and therefore exhibit an IR spectrum. Rotational transitions often occur simultaneously with vibrational transi tions. The combined vibrational-rotational energy relationship is of the form =
E
=
( 24)
where I is the moment of inertia of the molecule and m is a rotational quan tum number that can have values ± 1, ±2, and so on. The center of the vibrational-rotational band is at v0; thus the second term in Eq. (24)
210
Delgass and Wolf
defines t he rotational fine structure . Dependin g on w hethe r m i s positive or ne gative , two branches, known as P an d R br an c hes , are produced in the hi gh - and low - frequency sides of vo. Another branch , known as the Q branch, may app ear in t he vicinity of v 0 due to changes in angular momentum about the axis j oini n g the nuclei or d ue to unpaired electrons . Fi gure 24 s hows , in addition to adsorbed CO, t h e hi gh-resolution absorb ance spectrum of gaseous CO, depicting the R and P branches resolved into specific lines due to rotational transitions. Lower resistance spectra will sh ow only two broad bands about vo. For CO ad s orbe d on a m e tal s urface, the rotational fine structure disappears and give rise to a single band.
Poly atomic Molecules: Diatomic molecule s exhibit only one vibrational mode , but as more atom s ar e added to a molecule , additional vi br ations are p ossi ble . For a n onli n e ar poly atomic molecule co n t ai ning n atoms, 3n - 6 vibrations are possible ( 3n - 5 f or linear molecules). For these , n - 1 can be defined as stretching (opposite movement of atoms) and 2n - 5 are bending or deform ation vibrations ( atoms move at an an gle to each other ) . In some cases, the vibrations correspond to a similar type
90 r----.--�---.---�---.
78 66 0
Q
54
�
42
"
w
C( ro a: 0
� C(
30
-5 L---�----�---L--__J 2400 2250 2100 1950 1800 1650 1500 WAVE NUMBERS ( cm-1)
I n frared spectrum of CO in the gas plase and adsorbed on a c atalysts . T he d ata were collected at a resolution o f 2 em - 1 w ave number in a single scan in a F T I R instrument . T he fine structure superimposed into the broad peaks (P , R branches ) represent s rot ation al transitions . The sharp peak at 2070 cm- 1 cor resp on d s to adsorbed CO. FIGURE 2 4
5% Pt /Si0 2
Ca talytic Surfaces
211
of motion and can be distinguished only because they are perpendicular to eac h other ( degenerate vibration s ) . Hair ( 1 9 6 7 ) discu sses in detail t he number o f possible vibrations in polyatomic molecules .
Group Freq uencies: While any vibrational mode involves all atoms in a molecule, much of t he motion is dominated by a few atoms or group of atoms . Thus, whenever a characteristic combin ation of atoms. is p resent on a com pound , t he same char acteristic frequency appears more or less independent of the structur e of the rest of the molecule . Group frequencies permit one to identify struct ural g roups from t he IR s p ectra . Group frequencies often involve t erminal atoms of sm all mass such as C-H, O- H, N-H, or internal atom s connected by bonds which have a distinctive force constant . E xtensive tabulations and disc ussions regarding the use of group frequen cies for chemic al identification are available ( Alpert et al . , 1 97 0 ) . T he identification of the spectra of adsorbe d species often consists of matching t he sur face spectra with t hose of known compound s . Furthermore , infrared bands that are forbidden for molecules in the gas p hase on grounds of symmetry may be possible w hen t he molecule is adsorbed and p erturbed by surface forces . The frequency of the adsorbed s pecies shifts wit h surface coverage , co adsorption of other molecule s , particle si ze , or , in general , wit h chan ges in t he electronic environment o f t he metal-adsorbate interaction . B and Intensities: I f radiation of intensity I o imp inges on a media of thicknes s d containing a concentration of molecule s C , t he intensity of t he trans mitted radiation is given by the Beer-Lamb ert law, A =In
10 = ECd I
( 25)
where E is the extinction coefficient and A the absorbance. For adsorbed molecules , the extinction coefficient can be larger or smaller than in the gas p hase (Little , 1976). I f E is constant with coverage , there is a linear re lation between absorb ance and covera ge which can be used for quantitative analysis . T he best approac h to measuring extinction coefficients is t • the mean pore diameter , i s com puted as p
::
J"" 0
r f[ E: ( r ) ] dr p p p
�
( 8)
where f [ E:( rp ) ] drp is the fraction of the void volume with pores of radius + dr The tortuosity factor , ', accounts for the tortous path p of the molecules in the porous medium and the varying pore cross section . T his em pirical fac tor depends mainly on the porou s material , it s porosity , an d the method of preparation . The value of ' is us ually in the ran ge 2 to 7 . Wohlfahrt ( 1 98 2 ) presente d correlation s of ' for several types of porous cataly s t s . T he data for cataly sts prepared from compacted powders were c orrelated as rp to rp
•
( 9) where m dep end s on the porous m aterial . Typical data and values of the parameter m for several cataly sts are described in Fig . 1 . A d ditional data were report ed by Prob st an d Wholfahrt ( 1 97 9 ) . Feng and Stewart ( 197 3) pre dicted tha t ' s hould be about branches . It is reasonable to assume that are wide compared to the Stefan - Maxwell relations if the effec t iv e diffusion coefficien t s
3 for isotropic sy stem s with no dead - end that the fluxes in a porous media , with pores mean free path len gth , still satisfy the bin ary diffusion coefficient s are replaced by
( 1 0) Ornata and B rown ( 1 97 2 ) pointed out that the tort uosity factor is not nece s H oweve r , with sarily the same for both Knudsen an d molecular diffusion . out detailed measurements , this is a n ec essary assumption . Similarly , it is reasonable to expect that the forc ed flow throu gh a porous media satisfies the relation
N =
B oP � ll R T
dz
( 11)
242
1::. v
•
m
�.(
� �
0.70 V-oxide col. (S02 - oxid ) 0.85 NiO - col. unsintered
/
x 0.95 corbonyl - iron pellets. 5 p.m
+
L uss
�/ � ;{
1.05 porous Ag . 65 p.m
o 1.10 boehemi te , 90 p. m
0 1.60 porous boehmite 70p.m
/:
�#')
2 �n (E l"r )
�/
i
0
-I -2 -3 -4 -5
-
-6
/
A/ /
-
l
�n
-I
-2
-3
-7
( E /(2 - E ) )
FI G U R E 1
Dependence o f the tortuosity factor on the p or o si ty for several
materials .
( From Wohlfahrt ,
1 98 2 . )
w he re the p ermeability of the m ediu m , B o , depen ds on the geometry and structure of the pores . A method of fin din g the flux relation s in the intermediate situation s , where the three tran sport m ec hani sms m ay be of comparable importance , was p roposed by M a son and E vans ( 1 9 6 9 ) . T hey s uggested that the rate of m omentum transfe r by colli sion with the wall s an d a m on g molecules s hould be combined additively . This leads to a set of n in dependent equaD
tion s for the diffu sive fluxes Ni :
1 grad RT
N?
pi
1 = --
D
In the s peci a l case of
D
e
n
+ I:
ki
j=1 i=j
D
x.N. J 1
RT
ij e
( 12)
a bin ary mixture , Eq . ( 1 2 ) reduces to
( 1 3)
e A
D
D
x.N. 1 J
grad p
A
w he re
1 e D A
-
=
1
-- +
e
DK
A
1 - ( 1 + a) x
e D AB
A
( 14 )
24 3
Diffusion -React ion In teractions
an d =
a
( 1 5)
Solvin g th e n equation s E q . ( 1 2) gi v e s
N�
1 RT
=
1
n
G.
L .
where the elements of are 1]
F
1
.
.
G1J
are those of the matrix F
for
D l�.J +
1
--
=
ii
gra d p .
x.
=
F..
D
e
ki
n
-1
.
The entrie s of F
;t j ( 1 7)
x.
I: .....L D�.1]
j=1 i 4j
( 16)
1
.
lJ
J
for i =
M ason an d E vans a s s u m ed that the t o t a l fl u x is the su m of t h e diffu sive flux and viscous f low : N
i
=
N� 1
where
N:'1
=
+
-
N:'
( 18)
1
x B oP i
11 R T
( 1 9)
grad p
S u b stitution of E qs . ( 1 8 ) and ( 1 9 ) i n E q . ( 1 2 ) gives a relation fo r the total fluxe s :
N.
1 1 grad p = -- + e i RT
Dk
i
n
�
j=1
i :#j
x.N.1 ]
-
D�.
lJ
x.N . 1 ]
+
xB p i O
D k iJ.J R T e
gra d p
( 20)
An elec trical analog of the ad ditivity p rocedure p roposed by Mason an d E vans is shown in Fig . 2 . It is remarkable that identical additivity r ul e s are obtained by a ri gorous analysis b ased on the dusty - gas m odel (Mason et al . , 1 9 6 7 ) . Figure 3 enables a r apid ro u g h estimate of the dependence of the dif fusional regime on the average pore si ze and operatin g pressure . For example, when the pore size is ab ou t 1 0 0 A , K n u d sen flow is a dominan t me c h a ni s m for pressures b elow 5 x 1 0 5 Pa , while gaseous diffusion is the main contrib utor for p res s u re s above 5 x 10 6 Pa .
244
Luss
�� . �?+ �r (ADO FLOWS)
No
N' - I
- I
:s
a;_� -,N .., ..., ---
BULK DIFFUSION
+z
�
�
� !6l-s ---
dZ
c:...,- � ...,N
VISCOUS FLOW
KNUDSEN
DIFFUSION
DIFFUSIVE FLOW
FIG U RE 2 Electrical analog for the additivity rule s for the gaseous fluxes ( F rom Mason et al . , 1967. ) accordin g to the dusty - gas model .
10.000
... ...
o
