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Adsorption, Reaction and Desorption Rate Constants in Heterogeneous Catalysis, Measured Simultaneously by Gas Chromatography
M.1,. Occelli and R.G. Anthony (Editors ), Advances in Hydrotreating Catalysts 0 1989 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
211
ADSORPTION, REACTION AND DESORPTION RATE CONSTANTS INHETEROGENEOUS CATALYSIS, MEASURED SIEIULTANEOUSLY BY GAS CHROMATOGRAPHY
N.A.KATSANOS and J.KAPOLOS Physical Chemistry Laboratory, University of Patras, 26 110 Patras (Greece)
ABSTRACT Reversed-flow gas chromatography can be used to test solid catalysts with respect to their adsorptive and reactive properties, thus facilitating their design. The method uses a slightly modified gas chromatograph, by means of which diffusion bands are obtained when plotting the logarithm of the height of the extra "sample peaks", created by the flow reversals, as a function of time. In the presence of a catalyst, the diffusion bands are distorted because of slow rate processes and/or equilibrium states occuring in the catalyst's bed. Mathematical equations have been derived, by means of which the distorted diffusion bands are analyzed to yield rate constants, distribution coefficients, and overall mass transfer coefficients. INTRODUCTION The design of an effective catalyst should take into account, not only the reaction rate on its surface, but also the rate of adsorption of the reactant(s) and product(s) on the catalytic surface, as well as the rate of desorption of both reactant(s) and product(s) from the surface. The newly developed method of Reversed-flow Gas Chromatography (RF-GC) [1-3] can with advantage be used to test solid catalysts with respect to all above rates, measured simultaneously. From the rate constants of adsorption and desorption so determined, the distribution coefficients of the reactant and product between the catalyst and the gaseous phase, as well as the overall mass transfer coefficients in the gas phase and in the solid catalyst can be computed. Naturally, experiments at various temperatures can easily lead to activation energies and frequency factors, and also to heats and entropies of adsorption for the reactant and product on the catalyst's surface. EXPERIMENTAL The experimental set-up used with the RF-GC method is very
212
simple, consisting of a usual gas chromatograph equipped with a suitable detector for the vapor(s) of reactant(s) and product(s). The chromatograph is slightly modified as to include a usual four- or six-port valve, through which a chromatographic column is connected to the detector and the carrier gas, the latter being hydrogen in hydrogenation reactions. Other details in the experimental arrangement depend on the particular physicochemical quantity being measured. For the present purpose, the representation of the columns and gas connections is similar to that given elsewhere [ 4 ] and is shown in Fig.1. The sampling column I * + 1 the diffusion column 2 , and the lower vessel L ~ ,taken I
inlet of carrier g a s
-
reference injector
-
separation /column
detector
x.08
-'-aL+
x
xzf'
X=I:l
1
4
L
02
Fig.1. Outline of columns and gas connections in the RG-GC method for catalytic measurements.[4].
213
together, constitute what is called "the sampling cell". The branches l ' , 1 and L1 are constructed from ordinary 1/4 in. chromatographic tube and are usually 50-100 cm long. Vessel L2, containing the catalyst at its bottom, is wider (i.d. 15-20 m ) and has a volume of 2-10 cm 3 The whole sampling cell is accommodated inside the oven of the gas chromatograph with the branches l ' , 1 and L1 bent as ordinary chromatographic columns. The separation column can be placed in a separate oven and heated at a temperature different from that of the catalyst. Conditioning of the latter is carried out in situ with carrier gas (H2) flowing continuously through the sampling column. After some preliminary injections of the reactant (1-20 p1 of liquid or 1-3 cm3 of gas at atmospheric pressure) through the point z = L1 (cf. Fig.11, to establish constant catalytic activity, a fresh injection is made to study the kinetics for the various processes taking place on the catalytic surface. This is done by waiting for the first non-zero signal of the chromatographic detector in the recorder, and then reversing the flow-direction of the carrier gas for time t' = 10-60 s , by simply turning the four-port valve from one position (solid lines) to the other (dashed lines) and vice versa. The concentration c(l', to) of the various substances at x = 1 ' and at time to from the reactant injection, due to the diffusion of the various substance vapors along column z, is enriched by the flow reversal, this enrichment lasting only for a time period t'. In the absence of a separation column, an extra chromatographic peak, fairly symmetrical and narrow ("sample peak") would be obtained. Examples of sample peaks have been published many times [l-41. However, when more substances are present a x = l ' , e.g. a reactant and a product, this sample peak would be composite, comprising the extra concentrations of all substances created by the flow reversal. It is the purpose of the separation column, placed before the detector, to separate the various concentrations due to different substances, thus giving rise to more than one sample peaks, as exemplified by Fig.2. The procedure outlined above is repeated many times during a kinetic experiment, the sample peaks obtained each time representing a precise sampling with time of all substances present at the junction x = 1 ' .
.
MATHEMATICAL ANALYSIS If one plots lnh, where h
is the height of the sample peaks
214
I
36
35 i sample
,
win
U I 34
'iil
33
.
' D l L
2
32
peaks
time to/min Fig.2. Sample peaks of 1-butene (reactant, 1) and butane (product, 2) obtained during hydrogenation of the first over Pd/A1203 catalyst at 299.3.K, by reversing the flow-direction of the hydrogen carrier gas ( V = 0 . 3 3 C ~ ~ S for - ~ t' ) = 15 s , at to = 32 and 38 min after injection of 1 cm3 1-butene. The separation column was a 1.30 m x 1/8 in. chromosil 310 of Supelco SA. (measured from the ending baseline to their maximum) as a function of time to, a diffusion band is obtained. In the absence of catalyst, this band consists of a steep rise and a linear fall after the maximum. From the slope of this linear part, the diffusion coefficient into the carrier gas of the substance responsible for the sample peaks is easily calculated [3,41. In the presence of the catalyst, the diffusion bands are distorted, either in their shape or only in their slopes, and this is due to the slow rate processes and/or to equilibrium states occuring in the catalyst's bed. It is this distortion of the diffusion bands which permits the calculation of the various rate and equilibrium constants mentioned in the Introduction. This can be compared to the old way
215
of measuring physicochemical quantities by gas chromatography, based mainly on the distortion of a chromatographic elution band. Here, no chromatographic process pertains to the catalytic bed. Only a longitudinal diffusion current carring the effects of the various processes in the bed along the direction z, which is perpendicular to the carrier gas flow, and meets it at the junction x = Z'. In what follows the appropriate mathematical equations will be described or derived, by means of which the distorted diffusion band can be analyzed to yield the rate constants for adsorption of the reactant and product on the catalytic surface, the rate constant of the surface chemical reaction, and the rate constants for desorption of the reactant and product from the catalyst. From these primary physicochemical quantities, distribution coefficients and overall mass transfer coefficients of both reactant and product can also be computed. The Diffusion Band in the Absence of Catalyst The general mathematical equation describing a diffusion band, when no other process is taking place inside the sampling cell, is given by the following equation[5],intheform of its Laplace transform with respect to to:
V
where po) = transformed function of c(Z', to) with transform parameter po; m = amount of solute substance injected into the cell; = volumetric flow-rate of the carrier gas; = gaseous volumes of the diffusion column L1 and the VG' "G vessel L2, respectively; h = dimentionless transform parameter given by the relation C(Z',
+
rCLD
h = Po/P = Po/ -j-
L,I
D
= diffusion coefficient of the solute into the carrier
gas. The sample peak height is simply [1,31. h = 2c( Z', to)
(3)
216
and c(Z’, to) if found by taking the inverse Laplace transformation of eqn.1:
where
and rl, r2 are the roots of the denominator in eqn.1, which can be found with any desired precision, since the volumes VG and Vc of the cell are known by measurement. The two roots r1 and r2 have negative values, the one being at least 10 times absolutely bigger than the other. For instance, if vessel L2 is absent, V h = 0, and the roots are r1 = -3.073 and r2 = -0.2522. Thus, eqn.4, which describes the diffusion band after the maximum, is a sum of two exponential functions with negative exponents. That with the bigger root, say rl, becomes quickly of negligible value with time, leaving the function exp(r2Pto), which gives for the sample peak height lnh = ln(2N1
1 + r2 2
-
1
+ r2Pto
From the slope of the last linear part of the diffusion band, one then finds the value of r2P, and from this the diffusion coefficient D, since P = IT2 D/L:, according to eqn.2. The bigger the value of V;/VGI the smaller the absolute value of the root r2 becomes, slowing down the diffusion current along co-ordinate z (cf. Fig.1). The product -r2 p is thus an effective diffusion parameter peff, and is related to D by the equation
L
eff
where Leff = L1/ -r2
(8)
Diffusion Bands in the Presence of Catalyst If the volume of the catalyst placed at the bottom of vessel L2 (cf. Fig.1) is small compared with the gaseous volumes VG and Vh, the equivalent equation to eqn.1 is [ 5 ]
217
where p = po/Peff (cf. eqn.2), ki, k; and kI1 are dimensionless rate constants defined by the relations
and = rate constant for adsorption of the reactant on the cata-
kl
lyst surface;
k2
= pseudo-first order rate constant for chemical reaction
of the adsorbed reactant to give the adsorbed product, which is equal to k;CH2, C being the constant concentrationofH2 H2 adsorbed on the catalyst; = rate constant for desorption of the reactant from the surk-l face ; Peff = effective diffusion parameter defined by eqn.7. The diffusion band of the injected reactant comes out by taking the inverse transform of eqn.9:
where
X = 1 + n 2 k' 1
+ k; +
kll
2 2 (X2 - Y ) / 4 = k; + kll + n kik; Z =
X - 2(k;
+
kLl)
(14)
(15)
Equation 11 has the same appearance with eqn.30 of ref.6, although the meaning and the physical content of the parameters X, Y and Z are different. It describes a diffusion band distorted by the various phenomena occuring in the presence of the catalyst, and consists of the sum or difference of two exponential functions,
218
depending on the sign of the preexponential factors 1 + Z/Y and 1 -Z/Y. Coming now to the diffusion band of a reaction product, this presents two possibilities: either the pure product is injected (in the presence of catalyst) into the cell without being preceded by the reactant, or the band is due solely to the product as it is produced from the injected reactant. In the first case the band equation is again eqn.11 with N2 given by eqn.12, m and Peff now pertaining to the product. The parameters X I Y and 2 , with the subscript p denoting the product, are given by relations analogous to eqns.13, 14 and 15, with ki = 0 : X = 1 + P
TI
2 kip
+
2 - Y2 )/4 = k i (Xp
P
Zp - Xp
-
kllp IP
(16) (17)
2kllp
where
klp and kbeing the adsorption and desorption rate constant, 1P respectively, for the product on the catalyst surface, and PeffSp the effective diffusion parameter for the product, as defined by eqn.7 with the diffusion coefficient of the product D in place P of D. In the case that the product is not injected, but merely formed by the chemical reaction on the catalyst suriace, the mathematical function describing its diffusion band is derived here as follows. The substance is produced from the adsorbed concentration cs of the reactant, it desorbs from the catalyst surface and then diffuses away along column z towards the junction x = Z ' , from where it is carried to the detector giving sample peaks separated from those of the reactant (cf. Fig.2). It is the analytic function hp
2c ( Z ' ,
to) = 2f(to)
P
(19)
which is sought here. The diffusion equation (Fick's second law) for the product is ac /ato = D a 2c /az2
P
P
P
(20)
219
It can be solved by taking the to Laplace transforms of both sides, under the initial condition c ( z , O ) = 0, and subject to the bounP dry conditions at x = Z ' :
where v is the 1iner.velocity of the carrier gas and c (z', to) P the product concentration in the sampling column at x = 2 ' (cf. Fig.1). Then one obtains
D (dCp/dz)z,O = vC ( Z ' ,
P
P
pol
where capital letters, like C are used to denote the Laplace P' transformed functions. Equation 22 is an ordinary linear second-order equation, which can be easily integrated, either classically or by using z Laplace transformation. The second method leads to the following relation, taking into account also eqns 23: C = Cp( Z', po)cosh q z P P
+
VCP( L P o )
DPqP
sinh q z ' P
(24)
where L qp - PODP
(25)
The boundary condition at the other end z = Leff of the diffusion column is governed by the equation
where aG
K
SP
- cross-sectional area in regions -
x and z;
= overall mass transfer coefficient of the
product in the
solid ;
- total free surface area
of the catalyst; c = concentration of the product adsorbed on the solid catalyst; SP
As
220
c* SP
concentration of the adsorbed product in equilibrium with the gas phase concentration c P' The rate of change of c is given by the equation SP ac K A 3= 2 P - E (c* - csp) + k2cs SP =
vS
atO
where cs is the concentration of the adsorbed reactant, and Vs the total volume of the catalyst. The system of eqns.26 and 27 is treated by applying transformations with respect to to, under the initial condition c ( 0 ) = 0, and eliminating C between the SP SP transformed equations. To the result obtained, Cs (pertaining to the reactant) is substituted from a transformed equation similar to eqn.27: acs - KsAs atO
(c:
-
cS)
-
k2cs
vS
where c : is the equilibrium concentration of the adsorbed reactant and Ks its overall mass transfer coefficient in the solid. After this substitution and the use of a liner isotherm for the reactant:
K = c* S / c(Leff)
(29)
K being the distribution coefficient, there is obtained dC -D a (-1 p dz
z=Leff
= K A sp
Po Po
+
kip %p
is given where the desorption rate constant for the product k1P by the relation k-lp
K =*
A
vS
NOW, C* is substituted from a linear isotherm, analogous to SP eqn.29: K = c* /c (Leff) P SP P and then (dCp/dz)Z,L
(32) eff
and Cp(Leff) are found from eqn.24,whilst
221
C(Leff) is calculated from eqn.4 of ref.6 with Leff in place of L1. Using these three results in eq.30, one obtains, after taking v/Dq > > 1 and v/D q > > 1, because of the high enough flowP P rate of the carrier gas:
CP(Z',PO)
=
c(Z',po) (K;
k2k-lK/K )sinh qLeff
Dq(Po+k-,p) (P0+k2+k_l) [Dpqpsinh qpLeff (
coth qpLeff
DPqP where q2 = po/D, and K '
GP
~
"Ap
Dpqp 2 2
.
PO
Po+k-lp
(33)
is given by the relation
K
being the overall mass transfer coefficient of the product in GP the gas. Now, sinh qLeff, sinhqpLeff and coth qpLeff/Dpqp are approxi5 mated [51 by qLeff, qpLeff and (1/p0 + l/Peff.p)/Lefff respectively, giving Cp(l'rPo) z
(TI 2k'k'
k' KB 2/Sp)C(l',po)2 -1 Ip
where v , k;, kll have the same meaning as before (cf. eqn.101, referring,to the product, are defined as while B, k' and kl 1P 1P follows:
Finally, substitution in eqn.35 of the right-hand side of eqn.9 for C ( Z', po) gives
222 TI2mk;kllki
Cp(2',P0)
KB2
=
The function c ( Z ' , to) = f(to) is found by taking the P inverse Laplace transformation with respect to p, of eqn.39, which depends on the product's parameters kip (for adsorption) and kl 1P (for desorption), as well as on those of the reactant k;, kil and ki. The first bracket on the right-hand side of 39 is the same with the denominator of eqn.9 of the reactant, so the roots of this polynomial are given by
The roots of the second bracket of eqn.39 are
where X and Y have the same meaning as in eqns.16 and 17. P P Substituting the four roots above in 39, and inversing the transformation, one obtains
where N3 =
n2mkikllk; KBPeff OK P
=
N2
Q1 = -Y[X + Y -(X + Y )BI[X P P
n2kikllki KB2
KP
+ Y -(Xp - Yp)B1/4
Q2 = Y[X - Y -(X + Y )B][X-Y -(X P P P
-
Q3 = -Y B[X + Y -(X + Y )B][X-Y-(X P P P P
Yp)B1/4
+ YP )B]/4
(43)
223
CALCULATION OF RATE CONSTANTS AND OTHER COEFFICIENTS FROM EXPERIMENTAL DATA For the kinetics of a given reaction on a certain amount of catalyst, at one temperature T1, four experiments are basically required: (1) An injection of a small amount (cf. Experimental) of reactant into the sampling cell is made without the presence of catalyst. Then, reversals of the flow direction of the carrier gas are performed for a constant short time interval, noting the time to when each reversal is made, as measured from the moment of the injection. The height h (in arbitrary units, say cm) of the sample peaks resulting from the flow reversals is measured as shown in Fig.2, and the diffusion band is constructed by plotting lnh versus to. An example for the band of such an experiment is given by Curve 1 in Fig.3. (2) The same experiment without catalyst as in ( 1 ) is repeated with the pure product (cf. Curve 2 in Fig.3). (3) After placing a known weight of catalyst at the bottom of vessel L2 of the same cell, conditioning the catalytic bed, etc. (cf. Experimental), an experiment like (11 is conducted with the reactant, each flow reversal being repeated after the recording of all sample peaks for reactant and product(s) due to the preceding reversal. A separate diffusion band is constructed for each substance, i.e. for each kind of sample peaks (cf. Curves 3 and 4 in Fig.3). (4) When the height of the sample peaks in the previous experiment has been decayed to a negligibly low detector signal, pure product is injeted and the experiment described in (2) is repeated in the presence of catalyst (cf. Curve 5 in Fig.3). The slope of the last linear part after the maximum of the diffusion bands resulting from the experiments ( 1 ) and ( 2 ) gives r2P = -Peff and r p = - PeffSp, respectively, at the temperature 2 P TI, according to eqn.6. The value of Leff for the cell is calculated from its volumes VG and Vh, without any kinetic experiment, by simply solving the quadratic equation (cf. eqn.1):
224
%
10
a-
'0,
t 9 -
n
i8 1,
8
E
-?
4 - 7 E:
M
6
0
100
200
t,/min
Fig.3. Diffusion bands of 1-butene and butane obtained at 403 K with a sampling cell of VG = 6.42 cm3, VG = 13.533 cm3 and L1 = 78 cm. Curve 1 : 1 cm3 of 1-butene injected without catalyst; curve 2 : 1 cm3 of butane injected without catalyst; curve 3 : 1 cm3 of 1-butene injected in the presence of 461 mg of 60% Ni/Al203 catalyst; curve 4 : butane obtained from the reaction of the injected 1-butene on the same catalyst as in curve 3; curve 5 : 1 cm3 of butane injected in the presence of the same catalyst as in curve 3 ; The carrier gas was pure H2 with a volume flow rate of 0.25 cm3s-1
.
(1.29
+ n 2Vi/VG)A2
t (4.29
+
n2V'/C G G) h
+
1 = 0
225
The smaller root r2 is used to calculate Leff by means of eqn.8.. From the distorted diffusion band of the reactant obtained in experiment (3), the two exponential coefficients (X + Y)peff/2 and ( X - Y)peff/2, and the two respective pre-exponential factors N2(1 + Z/Y)/2 and N2(1 - Z/Y)/2 are computed. This is done either by using a suitable computer program (non-linear regression analysis), or, if the last part after the maximum is linear, by finding the slope of this, say -(X -Y)peff/2 and the intercept In" 2 (1 - Z/Y)], and then reploting the initial data before the maximum as ln{h -N2(1- Z/Y)exp[-(X-Y)Peffto/21} versus to to find -(X+Y)peff/2 from the slope of the new straight line obtained, and ln[N2(1+Z/Y)I from its intercept. Having found the values of the exponential coefficients (XtY)peff/2 and (X-Y)peff/2, and the respective pre-exponential factors N2(1+Z/Y) and N2(1-Z/Y), it needs only simple arithmetic to calculate X , Y and Z, and from them the rate constants kl, k2 and k-l €or the reactant. For instance, addition of the two exponential coefficients and then division of their sum by Peff (found from experiment I) gives the value of X. Subtraction of the same coefficients and then division by Peff yields Y. Finally, from the ratio p of the two pre-exponential factors, one finds p = -
1 -Z/Y 1 + Z/Y
and from this
z=-1-P
(49)
l+P
The fact that arbitrary units are used for the height h of the sample peaks, from which a diffusion band is constructed (cf. p.13) does not influence the value of Z , since it is calculated from the ratio p of two intercepts pertaining to the same substance and to the same experiment, so that any unknown proportionality factors cancel out. The values of X, Y and Z are now used in conjunction with eqns.13, 14 and 15. According to this relations k;
= (X+Z
ki =
-2)/2n2
x2 - Y2 - 2 (X - Y) 2(X + Y) -4
(50)
(51)
226
and from these dimensionless rate constants, kl, k2 and k-l in s-1 are found by multiplication with Peff (cf. eqn.190). An alternative way 5[ 5 ] , without using the values of the preexponential factors, is to conduct two experiments at the same temperature with two different lengths Leff. Coming now to the calculation of the other physicochemical parameters, the distribution constant K and the overall mass transfer coefficients in the gas and in the solid phase KG and Ks, respectively, for the reactant are found using the relations: kl =
KGAs/aGLeff
k-l = KsAs/Vs K = KG/Ks An analogous procedure is used to determine klp, k-lp, KGp, K and K for the product from the results of experiment (4). SP P From the exponential coefficients (X +Yp)Peffep/2 and P p'( - 'p)Peff.p /2 of the product, using eqns.16 and 17, one finds from the product II of these coefficients
and from their sum
C
After that, K Ksp and K are easily calculated using eqns.37 GP' P and 38, and also the relation K = K /Ksp, all these being equiP GP valent to eqns.53, 54 and 55, for the product. Finally, a crucial confirmation for the parameters determined is to use their values to calculate the right-hand side of eqn.42, since X, Y, Xp, Peff' Peff.p are all known. The coefficient N3 is calculated using eqn.43 and the value N2 found from the two pre-exponential factors in experiment ( 3 ) . The simple addition of these two factors gives 2N2. The calculated diffusion band can then be compared with the actual experimental one obtained from the product sample peaks in experiment ( 3 ) . The factor 2 in eqn.3 must always be kept in mind.
227
TWO LIMITING CASES OF THE EQUATIONS DERIVED
If the distribution coefficient K or K has a high value, meaP ning a small value of the desorption rate constant k-l or kIP compared to the respective adsorption rate constants kl and k IP the concentration of the reactant and/or the product reaching the junction x = I' (cf. Fig.1) may be very low and the sample peaks recorded may have negligible height. If this happens only with the product, no parameter pertaining to this substance can be determined, but the rate constants kl, k-l, k2, the distribution constant K and the mass transfer coefficients KG and Ks for the reactant are normally measured, as already described, without being influenced. Experiments (2) and (4) are not needed in this case. An example belonging to this category is offered by the action of sulfur dioxide gas on marble, when the product calcium sulfate does not desorb from the solid. If the reactant does not desorb, but the product does, eqn.11 cannot be applied, but eqn.42 can, and using the values of X and P Y determined from experiment ( 4 1 , the coefficients (X+Y)peff/2 PI and (X-Y)peff/2 can be calculated using a suitable computer program. Ther, omitting kll from eqns.13 and 14, one obtains X = 1
+ nLk; + ki
(58
Y = 1
iIT
2k ' 1
(59
ki
meaning that the coefficient (X+Y)peff/2 equals(l+n2ki)peff, while (X-Y)peff/2 is equal to kiPeff, from which kl and k2 are easily found. In the limiting case described above all experiments (1)-(4) are necessary. An example of this case is the dehydration of a higher alcohol over an alumina catalyst yielding alkenes. REFERENCES 1 N.A.Katsanos and G.Karaiskakis, Adv.Chromatogr., 24(1984) 125-180. 2 N.A.Katsanos and G.Karaiskakis, Analyst, 112 (1987) 809-813. 3 N.A.Katsanos, Flow Perturbation Gas Chromatography, M.Dekker, New York, 1988. 4 N.A.Katsanos, J.Chromatogr., 446 (1988) 39-53. 5 J.Kapolos, N.A.Katsanos and A.Niotis, Chromatographia, submitted for publication. 6 N.A.Katsanos, P.Agathonos and A.Niotis, J.Phys.Chem., 92 (1988) 1645-1650.
211
ADSORPTION, REACTION AND DESORPTION RATE CONSTANTS INHETEROGENEOUS CATALYSIS, MEASURED SIEIULTANEOUSLY BY GAS CHROMATOGRAPHY
N.A.KATSANOS and J.KAPOLOS Physical Chemistry Laboratory, University of Patras, 26 110 Patras (Greece)
ABSTRACT Reversed-flow gas chromatography can be used to test solid catalysts with respect to their adsorptive and reactive properties, thus facilitating their design. The method uses a slightly modified gas chromatograph, by means of which diffusion bands are obtained when plotting the logarithm of the height of the extra "sample peaks", created by the flow reversals, as a function of time. In the presence of a catalyst, the diffusion bands are distorted because of slow rate processes and/or equilibrium states occuring in the catalyst's bed. Mathematical equations have been derived, by means of which the distorted diffusion bands are analyzed to yield rate constants, distribution coefficients, and overall mass transfer coefficients. INTRODUCTION The design of an effective catalyst should take into account, not only the reaction rate on its surface, but also the rate of adsorption of the reactant(s) and product(s) on the catalytic surface, as well as the rate of desorption of both reactant(s) and product(s) from the surface. The newly developed method of Reversed-flow Gas Chromatography (RF-GC) [1-3] can with advantage be used to test solid catalysts with respect to all above rates, measured simultaneously. From the rate constants of adsorption and desorption so determined, the distribution coefficients of the reactant and product between the catalyst and the gaseous phase, as well as the overall mass transfer coefficients in the gas phase and in the solid catalyst can be computed. Naturally, experiments at various temperatures can easily lead to activation energies and frequency factors, and also to heats and entropies of adsorption for the reactant and product on the catalyst's surface. EXPERIMENTAL The experimental set-up used with the RF-GC method is very
212
simple, consisting of a usual gas chromatograph equipped with a suitable detector for the vapor(s) of reactant(s) and product(s). The chromatograph is slightly modified as to include a usual four- or six-port valve, through which a chromatographic column is connected to the detector and the carrier gas, the latter being hydrogen in hydrogenation reactions. Other details in the experimental arrangement depend on the particular physicochemical quantity being measured. For the present purpose, the representation of the columns and gas connections is similar to that given elsewhere [ 4 ] and is shown in Fig.1. The sampling column I * + 1 the diffusion column 2 , and the lower vessel L ~ ,taken I
inlet of carrier g a s
-
reference injector
-
separation /column
detector
x.08
-'-aL+
x
xzf'
X=I:l
1
4
L
02
Fig.1. Outline of columns and gas connections in the RG-GC method for catalytic measurements.[4].
213
together, constitute what is called "the sampling cell". The branches l ' , 1 and L1 are constructed from ordinary 1/4 in. chromatographic tube and are usually 50-100 cm long. Vessel L2, containing the catalyst at its bottom, is wider (i.d. 15-20 m ) and has a volume of 2-10 cm 3 The whole sampling cell is accommodated inside the oven of the gas chromatograph with the branches l ' , 1 and L1 bent as ordinary chromatographic columns. The separation column can be placed in a separate oven and heated at a temperature different from that of the catalyst. Conditioning of the latter is carried out in situ with carrier gas (H2) flowing continuously through the sampling column. After some preliminary injections of the reactant (1-20 p1 of liquid or 1-3 cm3 of gas at atmospheric pressure) through the point z = L1 (cf. Fig.11, to establish constant catalytic activity, a fresh injection is made to study the kinetics for the various processes taking place on the catalytic surface. This is done by waiting for the first non-zero signal of the chromatographic detector in the recorder, and then reversing the flow-direction of the carrier gas for time t' = 10-60 s , by simply turning the four-port valve from one position (solid lines) to the other (dashed lines) and vice versa. The concentration c(l', to) of the various substances at x = 1 ' and at time to from the reactant injection, due to the diffusion of the various substance vapors along column z, is enriched by the flow reversal, this enrichment lasting only for a time period t'. In the absence of a separation column, an extra chromatographic peak, fairly symmetrical and narrow ("sample peak") would be obtained. Examples of sample peaks have been published many times [l-41. However, when more substances are present a x = l ' , e.g. a reactant and a product, this sample peak would be composite, comprising the extra concentrations of all substances created by the flow reversal. It is the purpose of the separation column, placed before the detector, to separate the various concentrations due to different substances, thus giving rise to more than one sample peaks, as exemplified by Fig.2. The procedure outlined above is repeated many times during a kinetic experiment, the sample peaks obtained each time representing a precise sampling with time of all substances present at the junction x = 1 ' .
.
MATHEMATICAL ANALYSIS If one plots lnh, where h
is the height of the sample peaks
214
I
36
35 i sample
,
win
U I 34
'iil
33
.
' D l L
2
32
peaks
time to/min Fig.2. Sample peaks of 1-butene (reactant, 1) and butane (product, 2) obtained during hydrogenation of the first over Pd/A1203 catalyst at 299.3.K, by reversing the flow-direction of the hydrogen carrier gas ( V = 0 . 3 3 C ~ ~ S for - ~ t' ) = 15 s , at to = 32 and 38 min after injection of 1 cm3 1-butene. The separation column was a 1.30 m x 1/8 in. chromosil 310 of Supelco SA. (measured from the ending baseline to their maximum) as a function of time to, a diffusion band is obtained. In the absence of catalyst, this band consists of a steep rise and a linear fall after the maximum. From the slope of this linear part, the diffusion coefficient into the carrier gas of the substance responsible for the sample peaks is easily calculated [3,41. In the presence of the catalyst, the diffusion bands are distorted, either in their shape or only in their slopes, and this is due to the slow rate processes and/or to equilibrium states occuring in the catalyst's bed. It is this distortion of the diffusion bands which permits the calculation of the various rate and equilibrium constants mentioned in the Introduction. This can be compared to the old way
215
of measuring physicochemical quantities by gas chromatography, based mainly on the distortion of a chromatographic elution band. Here, no chromatographic process pertains to the catalytic bed. Only a longitudinal diffusion current carring the effects of the various processes in the bed along the direction z, which is perpendicular to the carrier gas flow, and meets it at the junction x = Z'. In what follows the appropriate mathematical equations will be described or derived, by means of which the distorted diffusion band can be analyzed to yield the rate constants for adsorption of the reactant and product on the catalytic surface, the rate constant of the surface chemical reaction, and the rate constants for desorption of the reactant and product from the catalyst. From these primary physicochemical quantities, distribution coefficients and overall mass transfer coefficients of both reactant and product can also be computed. The Diffusion Band in the Absence of Catalyst The general mathematical equation describing a diffusion band, when no other process is taking place inside the sampling cell, is given by the following equation[5],intheform of its Laplace transform with respect to to:
V
where po) = transformed function of c(Z', to) with transform parameter po; m = amount of solute substance injected into the cell; = volumetric flow-rate of the carrier gas; = gaseous volumes of the diffusion column L1 and the VG' "G vessel L2, respectively; h = dimentionless transform parameter given by the relation C(Z',
+
rCLD
h = Po/P = Po/ -j-
L,I
D
= diffusion coefficient of the solute into the carrier
gas. The sample peak height is simply [1,31. h = 2c( Z', to)
(3)
216
and c(Z’, to) if found by taking the inverse Laplace transformation of eqn.1:
where
and rl, r2 are the roots of the denominator in eqn.1, which can be found with any desired precision, since the volumes VG and Vc of the cell are known by measurement. The two roots r1 and r2 have negative values, the one being at least 10 times absolutely bigger than the other. For instance, if vessel L2 is absent, V h = 0, and the roots are r1 = -3.073 and r2 = -0.2522. Thus, eqn.4, which describes the diffusion band after the maximum, is a sum of two exponential functions with negative exponents. That with the bigger root, say rl, becomes quickly of negligible value with time, leaving the function exp(r2Pto), which gives for the sample peak height lnh = ln(2N1
1 + r2 2
-
1
+ r2Pto
From the slope of the last linear part of the diffusion band, one then finds the value of r2P, and from this the diffusion coefficient D, since P = IT2 D/L:, according to eqn.2. The bigger the value of V;/VGI the smaller the absolute value of the root r2 becomes, slowing down the diffusion current along co-ordinate z (cf. Fig.1). The product -r2 p is thus an effective diffusion parameter peff, and is related to D by the equation
L
eff
where Leff = L1/ -r2
(8)
Diffusion Bands in the Presence of Catalyst If the volume of the catalyst placed at the bottom of vessel L2 (cf. Fig.1) is small compared with the gaseous volumes VG and Vh, the equivalent equation to eqn.1 is [ 5 ]
217
where p = po/Peff (cf. eqn.2), ki, k; and kI1 are dimensionless rate constants defined by the relations
and = rate constant for adsorption of the reactant on the cata-
kl
lyst surface;
k2
= pseudo-first order rate constant for chemical reaction
of the adsorbed reactant to give the adsorbed product, which is equal to k;CH2, C being the constant concentrationofH2 H2 adsorbed on the catalyst; = rate constant for desorption of the reactant from the surk-l face ; Peff = effective diffusion parameter defined by eqn.7. The diffusion band of the injected reactant comes out by taking the inverse transform of eqn.9:
where
X = 1 + n 2 k' 1
+ k; +
kll
2 2 (X2 - Y ) / 4 = k; + kll + n kik; Z =
X - 2(k;
+
kLl)
(14)
(15)
Equation 11 has the same appearance with eqn.30 of ref.6, although the meaning and the physical content of the parameters X, Y and Z are different. It describes a diffusion band distorted by the various phenomena occuring in the presence of the catalyst, and consists of the sum or difference of two exponential functions,
218
depending on the sign of the preexponential factors 1 + Z/Y and 1 -Z/Y. Coming now to the diffusion band of a reaction product, this presents two possibilities: either the pure product is injected (in the presence of catalyst) into the cell without being preceded by the reactant, or the band is due solely to the product as it is produced from the injected reactant. In the first case the band equation is again eqn.11 with N2 given by eqn.12, m and Peff now pertaining to the product. The parameters X I Y and 2 , with the subscript p denoting the product, are given by relations analogous to eqns.13, 14 and 15, with ki = 0 : X = 1 + P
TI
2 kip
+
2 - Y2 )/4 = k i (Xp
P
Zp - Xp
-
kllp IP
(16) (17)
2kllp
where
klp and kbeing the adsorption and desorption rate constant, 1P respectively, for the product on the catalyst surface, and PeffSp the effective diffusion parameter for the product, as defined by eqn.7 with the diffusion coefficient of the product D in place P of D. In the case that the product is not injected, but merely formed by the chemical reaction on the catalyst suriace, the mathematical function describing its diffusion band is derived here as follows. The substance is produced from the adsorbed concentration cs of the reactant, it desorbs from the catalyst surface and then diffuses away along column z towards the junction x = Z ' , from where it is carried to the detector giving sample peaks separated from those of the reactant (cf. Fig.2). It is the analytic function hp
2c ( Z ' ,
to) = 2f(to)
P
(19)
which is sought here. The diffusion equation (Fick's second law) for the product is ac /ato = D a 2c /az2
P
P
P
(20)
219
It can be solved by taking the to Laplace transforms of both sides, under the initial condition c ( z , O ) = 0, and subject to the bounP dry conditions at x = Z ' :
where v is the 1iner.velocity of the carrier gas and c (z', to) P the product concentration in the sampling column at x = 2 ' (cf. Fig.1). Then one obtains
D (dCp/dz)z,O = vC ( Z ' ,
P
P
pol
where capital letters, like C are used to denote the Laplace P' transformed functions. Equation 22 is an ordinary linear second-order equation, which can be easily integrated, either classically or by using z Laplace transformation. The second method leads to the following relation, taking into account also eqns 23: C = Cp( Z', po)cosh q z P P
+
VCP( L P o )
DPqP
sinh q z ' P
(24)
where L qp - PODP
(25)
The boundary condition at the other end z = Leff of the diffusion column is governed by the equation
where aG
K
SP
- cross-sectional area in regions -
x and z;
= overall mass transfer coefficient of the
product in the
solid ;
- total free surface area
of the catalyst; c = concentration of the product adsorbed on the solid catalyst; SP
As
220
c* SP
concentration of the adsorbed product in equilibrium with the gas phase concentration c P' The rate of change of c is given by the equation SP ac K A 3= 2 P - E (c* - csp) + k2cs SP =
vS
atO
where cs is the concentration of the adsorbed reactant, and Vs the total volume of the catalyst. The system of eqns.26 and 27 is treated by applying transformations with respect to to, under the initial condition c ( 0 ) = 0, and eliminating C between the SP SP transformed equations. To the result obtained, Cs (pertaining to the reactant) is substituted from a transformed equation similar to eqn.27: acs - KsAs atO
(c:
-
cS)
-
k2cs
vS
where c : is the equilibrium concentration of the adsorbed reactant and Ks its overall mass transfer coefficient in the solid. After this substitution and the use of a liner isotherm for the reactant:
K = c* S / c(Leff)
(29)
K being the distribution coefficient, there is obtained dC -D a (-1 p dz
z=Leff
= K A sp
Po Po
+
kip %p
is given where the desorption rate constant for the product k1P by the relation k-lp
K =*
A
vS
NOW, C* is substituted from a linear isotherm, analogous to SP eqn.29: K = c* /c (Leff) P SP P and then (dCp/dz)Z,L
(32) eff
and Cp(Leff) are found from eqn.24,whilst
221
C(Leff) is calculated from eqn.4 of ref.6 with Leff in place of L1. Using these three results in eq.30, one obtains, after taking v/Dq > > 1 and v/D q > > 1, because of the high enough flowP P rate of the carrier gas:
CP(Z',PO)
=
c(Z',po) (K;
k2k-lK/K )sinh qLeff
Dq(Po+k-,p) (P0+k2+k_l) [Dpqpsinh qpLeff (
coth qpLeff
DPqP where q2 = po/D, and K '
GP
~
"Ap
Dpqp 2 2
.
PO
Po+k-lp
(33)
is given by the relation
K
being the overall mass transfer coefficient of the product in GP the gas. Now, sinh qLeff, sinhqpLeff and coth qpLeff/Dpqp are approxi5 mated [51 by qLeff, qpLeff and (1/p0 + l/Peff.p)/Lefff respectively, giving Cp(l'rPo) z
(TI 2k'k'
k' KB 2/Sp)C(l',po)2 -1 Ip
where v , k;, kll have the same meaning as before (cf. eqn.101, referring,to the product, are defined as while B, k' and kl 1P 1P follows:
Finally, substitution in eqn.35 of the right-hand side of eqn.9 for C ( Z', po) gives
222 TI2mk;kllki
Cp(2',P0)
KB2
=
The function c ( Z ' , to) = f(to) is found by taking the P inverse Laplace transformation with respect to p, of eqn.39, which depends on the product's parameters kip (for adsorption) and kl 1P (for desorption), as well as on those of the reactant k;, kil and ki. The first bracket on the right-hand side of 39 is the same with the denominator of eqn.9 of the reactant, so the roots of this polynomial are given by
The roots of the second bracket of eqn.39 are
where X and Y have the same meaning as in eqns.16 and 17. P P Substituting the four roots above in 39, and inversing the transformation, one obtains
where N3 =
n2mkikllk; KBPeff OK P
=
N2
Q1 = -Y[X + Y -(X + Y )BI[X P P
n2kikllki KB2
KP
+ Y -(Xp - Yp)B1/4
Q2 = Y[X - Y -(X + Y )B][X-Y -(X P P P
-
Q3 = -Y B[X + Y -(X + Y )B][X-Y-(X P P P P
Yp)B1/4
+ YP )B]/4
(43)
223
CALCULATION OF RATE CONSTANTS AND OTHER COEFFICIENTS FROM EXPERIMENTAL DATA For the kinetics of a given reaction on a certain amount of catalyst, at one temperature T1, four experiments are basically required: (1) An injection of a small amount (cf. Experimental) of reactant into the sampling cell is made without the presence of catalyst. Then, reversals of the flow direction of the carrier gas are performed for a constant short time interval, noting the time to when each reversal is made, as measured from the moment of the injection. The height h (in arbitrary units, say cm) of the sample peaks resulting from the flow reversals is measured as shown in Fig.2, and the diffusion band is constructed by plotting lnh versus to. An example for the band of such an experiment is given by Curve 1 in Fig.3. (2) The same experiment without catalyst as in ( 1 ) is repeated with the pure product (cf. Curve 2 in Fig.3). (3) After placing a known weight of catalyst at the bottom of vessel L2 of the same cell, conditioning the catalytic bed, etc. (cf. Experimental), an experiment like (11 is conducted with the reactant, each flow reversal being repeated after the recording of all sample peaks for reactant and product(s) due to the preceding reversal. A separate diffusion band is constructed for each substance, i.e. for each kind of sample peaks (cf. Curves 3 and 4 in Fig.3). (4) When the height of the sample peaks in the previous experiment has been decayed to a negligibly low detector signal, pure product is injeted and the experiment described in (2) is repeated in the presence of catalyst (cf. Curve 5 in Fig.3). The slope of the last linear part after the maximum of the diffusion bands resulting from the experiments ( 1 ) and ( 2 ) gives r2P = -Peff and r p = - PeffSp, respectively, at the temperature 2 P TI, according to eqn.6. The value of Leff for the cell is calculated from its volumes VG and Vh, without any kinetic experiment, by simply solving the quadratic equation (cf. eqn.1):
224
%
10
a-
'0,
t 9 -
n
i8 1,
8
E
-?
4 - 7 E:
M
6
0
100
200
t,/min
Fig.3. Diffusion bands of 1-butene and butane obtained at 403 K with a sampling cell of VG = 6.42 cm3, VG = 13.533 cm3 and L1 = 78 cm. Curve 1 : 1 cm3 of 1-butene injected without catalyst; curve 2 : 1 cm3 of butane injected without catalyst; curve 3 : 1 cm3 of 1-butene injected in the presence of 461 mg of 60% Ni/Al203 catalyst; curve 4 : butane obtained from the reaction of the injected 1-butene on the same catalyst as in curve 3; curve 5 : 1 cm3 of butane injected in the presence of the same catalyst as in curve 3 ; The carrier gas was pure H2 with a volume flow rate of 0.25 cm3s-1
.
(1.29
+ n 2Vi/VG)A2
t (4.29
+
n2V'/C G G) h
+
1 = 0
225
The smaller root r2 is used to calculate Leff by means of eqn.8.. From the distorted diffusion band of the reactant obtained in experiment (3), the two exponential coefficients (X + Y)peff/2 and ( X - Y)peff/2, and the two respective pre-exponential factors N2(1 + Z/Y)/2 and N2(1 - Z/Y)/2 are computed. This is done either by using a suitable computer program (non-linear regression analysis), or, if the last part after the maximum is linear, by finding the slope of this, say -(X -Y)peff/2 and the intercept In" 2 (1 - Z/Y)], and then reploting the initial data before the maximum as ln{h -N2(1- Z/Y)exp[-(X-Y)Peffto/21} versus to to find -(X+Y)peff/2 from the slope of the new straight line obtained, and ln[N2(1+Z/Y)I from its intercept. Having found the values of the exponential coefficients (XtY)peff/2 and (X-Y)peff/2, and the respective pre-exponential factors N2(1+Z/Y) and N2(1-Z/Y), it needs only simple arithmetic to calculate X , Y and Z, and from them the rate constants kl, k2 and k-l €or the reactant. For instance, addition of the two exponential coefficients and then division of their sum by Peff (found from experiment I) gives the value of X. Subtraction of the same coefficients and then division by Peff yields Y. Finally, from the ratio p of the two pre-exponential factors, one finds p = -
1 -Z/Y 1 + Z/Y
and from this
z=-1-P
(49)
l+P
The fact that arbitrary units are used for the height h of the sample peaks, from which a diffusion band is constructed (cf. p.13) does not influence the value of Z , since it is calculated from the ratio p of two intercepts pertaining to the same substance and to the same experiment, so that any unknown proportionality factors cancel out. The values of X, Y and Z are now used in conjunction with eqns.13, 14 and 15. According to this relations k;
= (X+Z
ki =
-2)/2n2
x2 - Y2 - 2 (X - Y) 2(X + Y) -4
(50)
(51)
226
and from these dimensionless rate constants, kl, k2 and k-l in s-1 are found by multiplication with Peff (cf. eqn.190). An alternative way 5[ 5 ] , without using the values of the preexponential factors, is to conduct two experiments at the same temperature with two different lengths Leff. Coming now to the calculation of the other physicochemical parameters, the distribution constant K and the overall mass transfer coefficients in the gas and in the solid phase KG and Ks, respectively, for the reactant are found using the relations: kl =
KGAs/aGLeff
k-l = KsAs/Vs K = KG/Ks An analogous procedure is used to determine klp, k-lp, KGp, K and K for the product from the results of experiment (4). SP P From the exponential coefficients (X +Yp)Peffep/2 and P p'( - 'p)Peff.p /2 of the product, using eqns.16 and 17, one finds from the product II of these coefficients
and from their sum
C
After that, K Ksp and K are easily calculated using eqns.37 GP' P and 38, and also the relation K = K /Ksp, all these being equiP GP valent to eqns.53, 54 and 55, for the product. Finally, a crucial confirmation for the parameters determined is to use their values to calculate the right-hand side of eqn.42, since X, Y, Xp, Peff' Peff.p are all known. The coefficient N3 is calculated using eqn.43 and the value N2 found from the two pre-exponential factors in experiment ( 3 ) . The simple addition of these two factors gives 2N2. The calculated diffusion band can then be compared with the actual experimental one obtained from the product sample peaks in experiment ( 3 ) . The factor 2 in eqn.3 must always be kept in mind.
227
TWO LIMITING CASES OF THE EQUATIONS DERIVED
If the distribution coefficient K or K has a high value, meaP ning a small value of the desorption rate constant k-l or kIP compared to the respective adsorption rate constants kl and k IP the concentration of the reactant and/or the product reaching the junction x = I' (cf. Fig.1) may be very low and the sample peaks recorded may have negligible height. If this happens only with the product, no parameter pertaining to this substance can be determined, but the rate constants kl, k-l, k2, the distribution constant K and the mass transfer coefficients KG and Ks for the reactant are normally measured, as already described, without being influenced. Experiments (2) and (4) are not needed in this case. An example belonging to this category is offered by the action of sulfur dioxide gas on marble, when the product calcium sulfate does not desorb from the solid. If the reactant does not desorb, but the product does, eqn.11 cannot be applied, but eqn.42 can, and using the values of X and P Y determined from experiment ( 4 1 , the coefficients (X+Y)peff/2 PI and (X-Y)peff/2 can be calculated using a suitable computer program. Ther, omitting kll from eqns.13 and 14, one obtains X = 1
+ nLk; + ki
(58
Y = 1
iIT
2k ' 1
(59
ki
meaning that the coefficient (X+Y)peff/2 equals(l+n2ki)peff, while (X-Y)peff/2 is equal to kiPeff, from which kl and k2 are easily found. In the limiting case described above all experiments (1)-(4) are necessary. An example of this case is the dehydration of a higher alcohol over an alumina catalyst yielding alkenes. REFERENCES 1 N.A.Katsanos and G.Karaiskakis, Adv.Chromatogr., 24(1984) 125-180. 2 N.A.Katsanos and G.Karaiskakis, Analyst, 112 (1987) 809-813. 3 N.A.Katsanos, Flow Perturbation Gas Chromatography, M.Dekker, New York, 1988. 4 N.A.Katsanos, J.Chromatogr., 446 (1988) 39-53. 5 J.Kapolos, N.A.Katsanos and A.Niotis, Chromatographia, submitted for publication. 6 N.A.Katsanos, P.Agathonos and A.Niotis, J.Phys.Chem., 92 (1988) 1645-1650.