lntraparticle Diffusion in Catalytic Heterogeneous Systems
EngFtring f?ocess development
AND NEAL R. AMUNDSON UNIVERSITY OF MINNESOTA, MINNEAPOLIS 14, MI".
NORMAN L. SMITH
This research was undertaken in order to determine the eEect of intraparticle diffusion in catalytic heterogeneous systems and to determine its relative importance on the over-all rate process. Formulas were derived which take into consideration diffusion of reactants and products, reaction rate, and mass transfer at the particle surface for three types of reactors: well-agitated continuousflow reactor, packed tube continuous reactor, and batch reactor. The reaction assumed that A F? B , a simple reversible reaction, first order in each direction, takes place inside the particle. The
equations obtained related the physical and chemical parameters of the system to the operating variables. In order to obtain experimental confirmation, the hydrolysis of an ester, ethyl formate, was studied using Dowex 50, a cation exchange resin, as a catalyst. Experiments performed on different sizes of particles for continuous flow and batch reactors, in general, validated the equations. The formulas should be of use in theoretical kinetic studies as well as in the design of commercial reactors. The experimental work showed that diffusion m a y be a factor in some heterogeneous processes.
I
back into the main body of the solution. It is possible also that there might be a resistance t o mass transfer a t the particle surface, and, for two of the cases, this will be taken into consideration. Thus if a reaction whose kinetics follows Equation 1 exists, it should be possible to consider all other kinetic effects which might arise. For a more complicated reaction mechanism some numerical scheme might be devised to yield results similar to those obtained here. Approximation methods, generally, do not offer conclusions from which generalizations may be drawn. Coupled with this difficulty is the tedium attendant on any numerical calculation. Modern computing machines may negate both these objections. The theoretical problems will be considered for three types of reactors.
N MANY chemical reactions in heterogeneous systems involving catalysis the diffusion of reactants into and products out of the catalyst particles influences the rate of product formation. The relation between the size of catalyst particles and catalytic activjlty had received some attention. Thiele (1%)concluded that for particles below a certain size the activity of a porous catalyst is proportional to the amount present and that for increasing grain size the activity depends on the total external surface of the grains. Following Thiele, other authors, notably, Wicke (15, 169, Wagner ( I d ) , Sanborn (IO),Psheehetskii and Rubinstein (9), Pshezhetskii (6-8),Zeldowitsch (18), and Dunoyer (2)considered various aspects of catalysis and its relation t o diffusion in both the steady and unsteady state. These writers have been concerned primarily with the effect of pore structure or with empirical correction factors for fixed bed operation. The present writers, on the other hand, have attempted to derive equations taking into consideration the diffusion of reactants and products and reaction rates in a catalyst particle for a simple reaction utilizing three different methods of carrying out the reaction. In order to facilitate the mathematical development the assumed reaction mechanism must be extremely simple. Certainly this restricts the applicability of the desired formulas, but there is some doubt whether general formalas may be derived for m y case which is more complicated. It is assumed that the reversible reaction
E "-A is described kinetically within the catalyst by the equation
--dCe dB
e
keCe - kaCa
where Caand C, are the concentrations of E and A , respectively, and IC, and k, are velocity constants for the forward and reverse reactions. respectively. Thus the velocity constants are characterized by the catalyst. It will be assumed that no volume changes occur, that diffusivities remain constant, and that the catalyst is in the form of uniform spheres of constant size, Reactant E must diffuse into the particle, form A , and then A must diffuse from the particle
1. Continuous-flow, agitated reactor 2. Fixed bed reactor 3. Batch reactor Case 1 is a steady state problem in which the catalyst is confined in a well-agitated reactor into which reactant is injected and product withdrawn continuously and is a special instance of the conventional stirred tank reactor. Case 2 is the usual packed tube reactor widely used in older industrial processes. Case 3 is the conventional laboratory method for making kinetic studies and is a transient problem and hence more complicated experimentally and theoretically. The essential assumption is made that there is no catalyst fouling or loss of activity. Efforts to take such factors into consideration lead t o almost insuperable difficulties. Once the equations had been derived a search was made for a system which would approximate to, a t least, all the hypotheses laid down in the theoretical development. The system decided upon was the hydro13 .is of an ester, ethyl formate, using Dowex 50, a cation exchange resin which exists in the form of nearly perfect spheres. This is not a porous catalyst but rather a gel-like medium, and it is felt that the diffusion81 effects should manifest themselves in an analogous manner. The theoretical equations for the three cases will be derived first, after which the experimental work will be described and application of the theoretical equations to the experiments will then be discussed.
2156
INDUSTRIAL AND ENGINEERING CHEMISTRY
September 1951
CASE 1: CONTINUOUS-FLOW, AGITATED REACTOR
A reactor contains W grams of catalyst and V cubic centimeters of solution. Catalyst diameter is R. The solution is admitted a t a rate of q cubic centimeters per minute having a concentration of E of moles per liter. Effluent solution is withdrawn at the same rate but with concentration C, in the in the reactant. If the feed solution contains product A and some A , say in a concentration E a t , it is clear because of the steady state that -
2157
In addition to these relations it is necessary to state the boundary conditions holding at the interface of the catalyst and the main body of the solution. These are
c.i
c,
c.i
+ c.i
=
E,
+ c,
It is assumed that the agitation is sufficient to reduce concentration gradients in the main body of the solution to zero. Hence the concentration of the effluent is the same as that of the solution in the reactor. Component E diffuses through a resisting film into the particle. Following this, diffusion must take place in the particle where reaction takes place to form A . Product A then diffuses from the particle through the resisting film and into the main body of the solution. The procedure of using an effective diffusivity in the particle will be followed here since i t is not certain that the pore structure of two supposedly identical batches of catalyst is the same. If one concentrates his attention on a single sphere and makes a material balance over a spherical shell of thickness Ar there results
-D,
(dg)
K,(C, -
=
Ea),when r
7-R
=
R
(7)
The quantities K , and K , are the mass transfer coefficients and are, in a sense, defined by these two equations. If there is no resistance to mass transfer-Le., if the agitation level is high enough-Equations 6 and 7 reduce to C,
=
??e,
-
when r = Iz
C, = C,, when r
=, R
(8)
(9)
by letting K , and K , become infinite. For the steady state operation of the reactor Equations 2 through 7 make up a complete mathematical description of the system. Although it is obvious physically, mathematically it is necessary to add the condition that concentrations inside the particles remain finite. In order to obtain a solution it is convenient to make a change of dependent variable ue = rC,
ua = rC, which reduces Equations 2 and 3 to
+
where r < ? < r Ar and De is the effective diffusivity of E in a catalyst whose fractional void volume,,is y. R , is the rate of disappearance of E as given by Equation 1. On letting Ar + 0 this equation reduces to The standard method of elimination can be used on these to obtain For product A , formed in the reaction, a similar analysis gives
D,
p$
+
rg)]+
k,C,
- k,C,
=
0
d4ue
d2u =-p-*= dr2 o
(3)
These two equations describe the relation between diffusion and chemical reaction within the particle. Consider now the relation between the concentration inside the particle and that in the main body of the solution. The rate a t u+ich E enters a single sphere is
where
p = - 'k+ " k De Da The solutions t o these are
and the rate at which it enters W grams of spheres is
*3
( C ) r= R
Hence this is the rate a t which the main body of the solution is being depleted in E by diffusion into the particles. If it is supposed that homogeneous catalysis can also take place in the main body of the solution by a mechanism similar to Equation l, E will disappear at a rate given by
From an over-all rate balance on E it follows that
where the eight constants appearing here are not all independent. Substitution of these two solutions into either of the second order differential equations shows that
ke
k, - a1 ka
Because of the finiteness required at r = 0, the solutions to Equations 2 and 3 are
C, = b,
A similar balance on A will produce
a2 =
b2 = k-, b,,
C, = Kb,
where e, = 2cl and K =
+
sinh r@
-L!
sinh
&, L = De D, k
T@
(10) (111
2158
Vol. 43, No. 9
INDUSTRIAL AND ENGINEERING CHEMISTRY
I n order to determine the values of bl and el it is necessary to go hack to Equations 4, 5, 6, and 7. I n particular, considering Equations 4 and 6 one sees that these involve C,, ??e, and c a a s unknowns with a known constant. Because the whole = and hence operation is in the steady state Ea can be eliminated. From Equations 4 and 6 a single relation can be derived
c,,,
ca ces+ cad ce
G
+ + 9,)
sinh ~ ( ( 1 g C 3(B,K BaL)}
=
+
+
w
+ + +
[(I - e , ) K (1 - e o ) L ] cash ~ [ ( l Q. ~ a ()K e e Len) 3(BeK BaL)] ( 1 7 ~ )
+
+
+
ICquntion 16 is the solution of the problem for the agitated reactor taking into consideration diffusion of reactant and product, reaction rate, mass transfer, and possible reaction in the main body of the solution. Considering the complexity of the problem originally proposed, the solution is relatively simple. As a spectial case one might consider the case in which resistance to Inass transfer is negligible, or e e = 0 = ea.
--
c, - Ea& =
where
c,, - -c,
=
- fe -- .fa - c,, tea.-
9
9
(1%
whei e
+ + Eieige - (1 + ga + QB)
+ ge + ga)bi A , - (1 + gs + Qa)bl - (1
,fa
= sinh w[g.(K
+ L) - 3 B , K ] + 3B,Kw cosh w
(1%)
+ L ) - 3B&] + 3BaLw cosh (19b) + gc + g a l ( K + L ) - 3(B.K + BaL)I + w
3(B,K
~ a ) ~=o
+ B,L)w cosh w
(1%)
With no reaction in the main body 6f the solution the formulas reduce to
If Equations 10 and 11 are substituted into Equations 12 and 13, respectively, one obtains Ai
= sinh
Q = sinh w I ( 1
From Equations 5 and 7 a similar relation can be obtained Cai(1
w[g,(K
fe
=
B i eRi
=
Bz R -
(14)
el
A further special case which will be considered in the experimental section is when 4 , = 0, k,, = 0, = 0 = e,. This is equivalent to K + so that
Ca -~- C a s -
(15)
C,t
with
ge
=
1
- 3Be + 3 B a d Z C o t h V Z - 3Be + 3Z?,l/Xcothl/&
+ 8.
(21)
where
+ 8.) + A Z = Cmge + (1 + B I = [(I + gB + go) (1 A1'=:Ze2(1
AI
QZa,
-
Qe)E'aa
cs)
- 3Belsinh w
Equation 21 can be written
+
+ Q. + g.)c + 3BeIw cash Bt = - [(l + g, + g.) (1 - ea)L - 3B,Llsinh w [(l+ + g a ) e a L + 3BaL], Gosh w [( 1
w
~e
where
Q
=
k,,V W -* 3
P
(1
-4%cothl/%)
Equation 22 is in a very useful form for experimental verification because of its linear dependence upon the flow rate, q, other parameters being constant. I n this formula it is assumed that the reaction E -+ A is irreversible and there are no mass transfer resistances.
Equations 14 and 15 can be solved for b , and e , to give
AZBI- BiAi
' - (1 + ga + gal ( B I K - Be)
b -
k , R2 De
CASE 2: CONTINUOUS-FLOW, PACKED TUBULAR REACTOR
With these values of the constants the concentrations C , and C , within the particle result. The concentration needed, of course, is Eawhich can be obtained by rearranging Equation 7
Performing all the needed operations, after much juggling, gives -
-
c,-c
(1%
- - F, - C-m - Fa - C 'e &- C e = C , , G G
(16)
whw?
F,
= sinhw(y,[(l
- ee)K + (1 w
F,
=
+
- ea)L] - 3 B e K ) Gosh w [ ~ e ( e e K 4) 3BeKI
+
+
-
+
+
sinh w { g e [ ( l - ee)K ( 1 . - ea)L] 3BaL) w posh w [ g a ( e e K eaL) 3BaLI
+
+
This problem has been considered more often in connection with industrial catalytic processes than almost any other, and the results obtained here are essentially generalizations of some results of Wagner. It is supposed that the catalyst is packed in a column and, for definiteness, operates upflow with distances along the reactor measured as 5 . All other quantities are then defined in the same way as before except q is the flow rate per square centimeter, W is the mass of spheres per cubic centimeter of column volume, and LY is the fraction voids in the bed. Equations 2 and 3 still describe the particle and Equations 6 and 7 hold a t the sphere surface. Equations 4 and 5 no longer hold but two new relationti can easily be derived to take their place. These are
(17a)
(17h)
- - 9 - = d- X
3wRyDa (2) + ( k 8 a C a - k,eE,)LY ,=R
(24)
INDUSTRIAL AND ENGINEERING CHEMISTRY
September 1951
and are obtained by isolating an element of the column length. These two relations are differential equations with indepmdent variable x , and hence two additional conditions are needed to dettxrmine the solution uniquely. These are -
C,
Ea = cai,when x
= 0
(25)
0
(26)
=
The complete mathematical description for this problem is given by Equations 2, 3, 6, 7, 23, 24, 25, and 26. Equ:it'ions 10 and 11 are still solutions of this system. The c,onstants el and bl can be redetermined using Equations 6 and 7. Substitution of Equations 10 and 11 into Equations 6 and 7, respectively, give
$ t
R
cosh
w
[ E ~ W
[ - caw cosh
+ (1 - ei) fiinh - (1 -
w
ea)
Thcse formulas should be useful because of the convenient form in which the various parameters are grouped. It should be noted that c e , f a , y,, ga, B e ,H a , and ware dimensionless quantities. C A S E 3: BATCH REACTOR
Gei,when 5
=
2159
W ]
sinh
=
W]
-
C,
=
-
hl
=
2 AI
-
C, - Kbl
where A I and A , are defined tiy thrse t~ivoequations. these equations results in
=
el -.
R
An
Solution of
The batch reactor, although it has been the favorite of most workers iri kinetics in the past, is-appreciahly more complicated from the theoretical point of view. This arises from the fact that the problem is a transient one, all concentrations varying with the time. For this reason the development will he Inad? herc for the irreversilrile reaction E -+ A only. It \vi11 also tie assumed that there is no resistance to mass transfer a t the catalyst surface. The latter restriction need not be made but, it, will be se(~nthat the solution is complicated enough without adding this gc.ncralization. Because the prohlcm is a transient on(' the tool for the solution is the Laplace transformation. In t,he batch reactor it is assumcci. that at zcro time thc iwxtor contains W grams of catalyst and T'cubic wntimcteru of solution. The concentrations inside the catalyst paiticlrs are given t1.v
b, = AIE, - LA&?,
C,
=
0, when t = 0
C,
=
0, when t = 0
~ p _ _
AiK - AZL
while t,he concentrations in the main body of thP solution itrc
e!
e,
= E e i , when t =
C,
=
- KC, - Z, R K A i - LA2 and these values may be mbstituted into Equation 10 or 11. S o w from = cai- E,, Equation 23 can be writ,ten
c. cei+
When Equation 10 is substitutc,d into this relation the following relation results
dC, -+-
F,
dx
+ Fa c
HR
e
P - -2(Cei HR
+
Fai,when
*itall times later-i.e., t > O-thc. surfacr \vi11 assumed t o he
c, c0
=
C,, when r
=
C,, when r
0
(34)
t =0
(35)
condition holding ut
tiicy
>0 = R, t > 0
=
R, t
sl)li(w~
(36) (37)
liquations 2 arid 3 are no longer valid inside the particlt. IJutit is a simplr: matter t,o show that in their place
Cai)
ivhere
and
H =
w
cosh w ( K e e
+ Leo) + sinh ~ [ (-l e,)K + (1
- ea)L]
and F , and Fa are as defined in Equations 17a and 17h. .I siniilar equation in E , ma?' be derived
must be substituted, The relations holding betm ecn particle concentrations and s o h tion concentrations can be derived in a manner similar to that of the previous two cases.
(28) The solutions to Equations 27 and 28 are, after making use of Equations 25 and 26,
For reasons wbich will he made clear later Equation 40 111 not be used but another term a ill be added (see experimrninl p : i i t )
In order to obtain the concentration Caas a function o f t l i v time it is necessary to solve the system of equations consistiiig o f 32 through 39, 41, and 42. Details of the Laplace tr;ttisfoim:tt ion method will not be prcwntod since they are availal,lta in : I v:Lri(lt.v of textbooks. The folloaing nohtion \rill be used The similarity between these formulas and those for the R ell-agitated reactor is rather striking. Many special cases of these two formulas can be written but one of interest is that for which kae = 0 , k,, = 0,E p = 0, ea = 0, = 0,
cai
-
C log&
c,,
=
3R -2 tanhw R tanh w
a formula obtained by lyagner.
w
m
L(C,!
=
1,(Ca)
=
h,
L(i2,)
=
ha
e - W e ( r , t)dt
(31)
=
he(r, p )
=
h,
INDUSTRIAL AND ENGINEERING CHEMISTRY
2160
The general procedure is t o apply the transformation t o the whole system thus obtaining a somewhat simpler mathematical problem. This problem is then solved t o obtain the transform of the desired solution, ha. It is necessary then to obtain the inverse of this which is I?,. The transform of Equation 38 is d2h, dT2
+ 2 ($) dh - ?+)
he = 0
Vol. 43, No. 9
previously and this can be done by taking the transform of Equation 41 to give
Once again it is to be noted that ha = ha when r = R so that on putting h. = U/P and performing the,operations indicated in this equation, A can be determined to be
The solution of this equation which remains finite a t r = 0 is
A r
he = - sin
rw
(43)
-
R
where
where A is a constant whose value must be found and
S(v) =
(v2
+ 3 8 ) sin v - 3pv cos v
To determine A the transform of Equation 42 is taken
Note that from Equation 36, Le = he when r = R, so that performing all the operations indicated in the above expmsion on Equation 43 gives an equation in A which may be solved for A to give
A = -R3
De (-w2
E
