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Homogeneous models for porous catalysts and tubular reactors with heterogeneous reactions
Chemical Engineering Science, 1975, Vol. 30, pp. 6X5-694.
Pergamon Press.
Printed in Great Britain
HOMOGENEOUS MODELS FOR POROUS CATALYSTS AND TUBULAR REACTORS WITH HETEROGENEOUS REACTIONS W. N. GILL, E. RUCKENSTEIN? and H. P. HSIEH Department of Chemical Engineering, State University of New York at Buffalo, Buffalo,NY 14214,U.S.A. (Receioed 5 September 1974;nccepted 6 Nooember 1974)
Abstractane-dimensional models for porous catalysts and laminar flow tubular reactors with first-order reactions at the walls are developed. The method which is used replaces the transport equations containing the local concentrations with equations, similar to those used in the dispersion theory, for the concentrations averaged over the cross section The main assumption is the approximation of the coefficients in the dispersion equations by those valid for a pulse of concentration introduced at time zero at the mouth of the pore or at the inlet of the tube. The effectiveness factor depends not only on the Thiele modulus but also on the length-to-radius ratio of the catalyst pore. The one-dimensional mode! for the catalytic tubular reactor provides a good approximation to the area-mean concentration for p (= k,RID) up to 1.0. The comparison is made with an exact orthogonal expansion solution developed also in the paper. For larger values of /3,say ,9 = 100,the discrepancies between the exact solution and the one-dimensional model may be as high as 50 per cent. Therefore, a generalized dispersion solution is presented to obtain more accurate predictions for fi > I .O. INTRODUCTION
After Tbiele [l] an approximate one-dimensional
model
d2C D;i;r=kC has been used widely[24] to describe diffusion of a reacting species in a cylindrical pore with a first order catalytic reaction at the wall. The “volumetric” rate constant, $, is related to the intrinsic rate constant per unit surface area, ks, by
$2
R
where R is the radius of the pore. In Eq. (1) one assumes that the concentrations at the wall and in the bulk are essentially the same. Bischoff [5] analyzed the same problem accounting, however, for the transverse diffusion. The heterogeneous reaction was treated as a boundary condition. One of the objectives of this paper is to propose a model having the simplicity of Thiele’s model, but which takes into account the concentration variation over the cross-section of the pore. The concentration averaged over the cross-section is the main quantity used in the equation. The method by which the new model is constructed is borrowed from the theory of dispersion processes[6]. The new model, which will be shown to produce results which agree well with BischotI’s results, indicates that the volumetric rate constant depends not only on k, and R as suggested by Eq. (2), but also on the pore diffusion coefficient D. The second goal of the paper is to apply the same procedure to more complex problems, for instance, to tTo whom all the correspondence should be addressed.
laminar flow in a tubular reactor with a wall-catalyzed reaction of the first order. The tubular flow reactor with catalyst deposited on the wall is useful for obtaining precise kinetic data from conversion measurements in view of its well-defined flow field[7]. Walker[8] suggested a one-dimensional model to describe the area-mean concentration in the tubular reactor: d2C,
dC,
kzdX2 + kldx
= kucm
where k, is given by Eq. (2), k, is equal to the negative of the average velocity and k2 is the molecular diffusion coefficient. Our goal is to develop an equation for the area-mean concentration of the same form, but without the assumption that the wall and the bulk concentrations are equal. The results obtained by this procedure will be compared with Walker’s result and with an exact solution. Although exact solutions are available[%12], a new solution is derived which has the advantage of computational ease and higher accuracy. The method which will be used to develop the new one-dimensional models for both the porous catalyst and the tubular reactor replaces the transport equation containing the local concentration with an equation for the concentration averaged over the cross-section. The equation is similar to those used in dispersion problems. The main assumption consists of approximating the coefficients in the dispersion equation by those valid for a pulse of concentration introduced at the initial moment at the mouth of the pore or at the inlet of the tube. The computation contains the following steps: (a) With a pulse (or a slug) introduced into a tube or a pore, the dispersion theory of Gill and Sankarasubramanian [6] expands the local concentration in
686
W. N. GILLetal.
series of the form (4) where Cr,,, is the area-mean concentration, X is the dimensionless axial coordinate and the functions X depend on the radial coordinate. (b) The area-mean concentration is formulated to follow the generalized dispersion equation
where r is a dimensionless time. It has been shown]131 that Eq. (5) can be truncated after the term involving KZ without introducing serious errors. The functions h and the coefficients Ki are related to each other and can be determined as shown in the Appendix. (c) The steady state values of K,, K, and Kt obtained for a pulse (or a slug) problem then are assumed to be directly applicable to steady state dispersion problems with constant inlet concentrations. That is, the coefficients K,,, K, and Kz, valid for a pulse, are used in the following equation to describe the steady state area-mean concentration in a pore or a tube: d?,,, dC,,, Kz+ K,z + KoC, = 0
Exact solution Bischoff [5] analyzed the above problem exactly and obtained for the local concentration
For the effectiveness factor, 7, he derived the equation:
(lot.4 . where 4 is the Thiele modulus defined by [3-51 (lla)
L = l/R.
(lib)
and
The PI’S are determined from the transcendental equation
(6)
with the constant inlet condition C(X = 0) = C,(X = 0) = c0. For large values of the catalytic parameter P(= k,R/D), Eq. (6) fails to describe tubular reactors with reactions on the wall accurately. A new result (Eq. 44) is given, which is relatively simple and accurate, for such systems.
Zk, “* eS=l m = LV(2B) ( >
pJdpi) = pI&i),
(7) The boundary conditions for a first-order chemical reaction at the wall are:
(12)
where ,3 ~ k RID s . NeW one_dimensional model The method outlined in the introduction gives, for systems without convection (see Appendix), KO=-pd
I.CYLINDRICALCATALYST PORES
Gooeming equations Consider a cylindrical pore of radius R and length 21 in a porous catalyst pellet. Neglecting any convection in the pore, the local concentration at steady state is described by
(i = 0, 1,2,3.. .I
K,=O
(13b)
K2= 1,
(13c)
and
where p. is the first (lowest in absolute magnitude) root of Eq. (12). The new model for cylindrical catalyst pores, therefore, becomes d2C, 3 = /.klV?tl
(14)
or, in terms of the dimensional coordinate x,
$(x,0) = 0
-+x,
R) = k,C(x, R)
(84 (W
d2C, _ D,x2-kbC,w The new volumetric rate constant, kd is given by
C(0, r) = Co
$1,r) = 0.
(15)
(16) (84
where p. depends on p.
Homogeneous models for porous catalysts and tubular reactors with heterogeneous reactions
The area-mean concentration, C,,,, has to satisfy the area-average forms of Eqs. (8~) and (8d). With the above developments, one obtains C L= co
cash [$‘(l -x)] cash (/&IR)
(17)
(b) Effectioeness factors. As a consequence of the dependence of k: on the diffusion coefficient, the new model predicts that the effectiveness factor can be significantly different from that obtained from the traditional model. The effectiveness factor computed from the traditional model is
and consequently the effectiveness factor is given by:
Discussion
(a) Volumetric rate constants. Comparison between Eqs. (1) and (15) indicates that a new volumetric rate constant k: should replace the traditional constant k,( = 2k,/R) in the one-dimensional models for cylindrical catalyst pores. The new constant depends on the diffusion coefficient D as well as on k, and R. The relation between these two constants has the form kl /LO -=k,
n=-
and is shown in Fig. I. As the wall catalytic parameter p approaches unity, the constant k. begins to deviate appreciably from the new constant kl (by nearly 30 per cent). For small values of p, Eq. (12) leads to pLo= v(2fi) and therefore kb- k,.
(2W
tanh 4 dJ ’
(21)
Both Bischoff’s result and the new model predict that the effectiveness factor depends not only on the Thiele modulus but also on the ratio L( = l/R). For large values of L Eq. (18) reduces to Eq. (21). The effectiveness factors obtained from the three approaches are given in Fig. 2. The differences between the results of the new model, Eq. (18), and those from Bischotf’s analysis, Eq. (lob), are fairly small over the range of practical interest. IO
(19)
2P
687
01
F 8 5 Y
0.01
8
Y ; B :: w
0.001
For large values of /3, pa = 2.4048 and ,+28916& P
“.
(2Ob)
These asymptotic results are included in Fig. 1. As expected, k, is close to k: when the reaction rate is slow. For instance, up to /? = 0.2, k, differs from k: by less than about 5 per cent. However, as /3 becomes larger, the ratio &L/k, becomes linear in l/p with a slope of 2.8916 and the two constants can differ by orders of magnitude. This happens, for example, when a relatively fast reaction occurs at the wall of a large pore in which a liquid reactant is diffusing.
00001
0.10
IO
Kl THIELE
MODULUS.
100
1000
#I
Fig. 2. Effectiveness factor vs Thiele modulus with dimensionless pore length as parameter. -, improved one-dimensional model, Eq. (18);---, Bischoff’s result, Eq. (lob).
However, the new model involves only one term and therefore is simpler to use than Bischotf’s expression. The figure shows that the traditional model may overestimate the effectiveness factor by orders of magnitude, especially when the pores are short in length and large in radius. This overestimation becomes serious when the Thiele modulus is large. Equation (1la) shows that, for a given finite pore length, /3 increases as the Thiele modulus 4 increases. When p becomes sufficiently huge, p. approaches 2.4048 and Eq. (18)givestheasymptoticresultconjectured byAris[21] ~ = (24048L) tanh (2.4048L) _ 1 6* P
(21a)
By contrast, the traditional model leads to
9--
Fig. 1. Ratio of the new to traditional volumetric rate constants vs the catalytic parameter 8.
for large p.
1
@lb)
W. N. GILL et al.
688
For many catalysts the length-to-radius ratios lie between 50 and 10’[5]. Therefore for gaseous reactants the effectiveness factor is close to the traditional predictions because they have small or moderate Thiele moduli. For liquid reactants, because of the low diffusion coefficient, the Thiele modulus is large, and therefore the new model is more appropriate than the traditional one.
IL WALL-CATALYZED TURULAR REACTORS
Describing equations
The procedure used in the previous section is applied to the case of laminar flow in a tubular reactor with first-order reactions at the wall. The reactant concentration in the tube of radius R is described by
are given by y”(y)
I A y2 e-*PY2M 2 - A.(1 + A.2/Pe2),,n ,
(
=
4
) (n = 0, 1,2,. . .)(25d)
where M(a, b, z) is the confluent hypergeometric function tabulated, for instance, in reference[l6]. Substituting Eq. (25d) into Eq. (25~) gives the following transcendental equation for the A.: 2A. - A:(1 + Az/Pe2)M 6- A,(1 + A~IPe’) 2 A , , n 4 2 2 - A.(I; A.‘/Pe’), 1, A. = o, > (n = 0, 1,2,. . J. (26)
+(p -A.)M
(
The coefficients B. from Eq. (24) can be calculated on the basis of the orthogonal properties of Y, ( y ) [9,17]. The final result is where u. is the centerline velocity. The concentration at the inlet is assumed to be constant, therefore, C(0, r) = CO.
(234
To complete the description of the system, the following boundary conditions are written:
With the local concentration given by Eq. (24) one obtains the area-mean concentration as m
$(x, 0) = 0
-+x,
R) = k,C(x, R)
~=2~~E”e-*:Xf’yY,(y)dy. 0
(23b)
(23~)
where k, is the heterogeneous rate constant. (a) Exact solution. Although exact solutions for Eqs. (22) and (23) are available[%12] another solution will be developed. Results are more accurate and easier to compute from this solution. The solution is formulated in terms of dimensionless axial and radial coordinates X and y as
(28)
(b) Two-dimensional plug flow model. Damkiihler [ 181 and Baron et al. [7] simplified .the problem by assuming a rod-like flow pattern with the velocity everywhere replaced by the average velocity of the flow. Their result accordingly becomes
oI exp{-y(J(lt$)-l)] %=4c
n=O
(29)
p1[1+(;)‘]
where the p.‘s are calculated from Eq. (12). -_ cs-2
B,e-A2XY.(y)
(24)
where A,,and Y.(y) can be shown to satisfy the relations ;gyF+Az
[
(Fy’)+$1Y" =o
$0) =0
W-4 (W
and
gql, =- pY"(l).
(25~)
It has been shownU51 that Y,, satisfying Eqs. (25aH25c)
(c) Previous one-dimensional models. Incontrast to the two-dimensional approaches described above the one dimensional models of the form of Eq. (6) for the area-mean concentration are simpler and hence easier to use. Walker[8] suggested a one-dimensional model with the coefficients K. = - 2p;
K, = - l/2;
KZ = l/Pe’.
(30)
Equation (6) with the coefficients K1 and K2 taken from the Taylor&is’ dispersion theory without reaction[l4,19] has been widely used in modeling homogeneous reactions in tubular reactors. Applying the same concept to heterogeneous reactions one obtains: K,,=-2p;
K,=-l/2;
Kz=1/192+1/Pe2.
(31)
689
Homogeneous models for porous catalysts and tubular reactors with heterogeneous reactions
(d) New one-dimensional model. As mentioned earlier, the coefficients I&, K, and X2 from Eqs. (A.24), (A.27) and (A.28) in the Appendix are valid for a pulse (or a slug) introduced into a pore or a tube with first order reactions at the wall. They are used to approximate the corresponding coefficients in the one-dimensional model for a step change at the inlet. Note that the coefficients in the new model differ from those given by Eq. (31) since the former takes into account the heterogeneous reactions at the wall. The concentration distribution from the onedimensional models, Eq. (6), can be shown to be Clll
Co=exp
_ K, + V(KIZ - 4&&)X 2K2 1.
(32)
It should be noted that the coefficient & is negative because reactant is consumed at the wall by the catalytic reaction.
that the centerline concentration at X =0*03 remains approximately the same for all p’s, while that at X = O-1 decreases appreciably as /3 increases. This is mainly due to dispersion, which is the combined action of axial convection and radial diffusion. For gas phase wall-catalyzed reactions the value of the catalytic parameter /3 is less than about 0.1[7,12]. The area-mean concentration distributions obtained from the new one-dimensional model, Eqs. (A.24), (A.27) and (A.28), are found to deviate from the exact solution by less than 3 per cent for the practical range of distances. For /3 SO.1 the new model gives results similar to those obtained by other models. For clarity of the presentation, in Fig. 4 only the new one-dimensional model and the two-dimensional plug flow model are compared with the exact solution.
Discussion Radial concentration profiles obtained from the exact solution, Eq. (24), are shown in Fig. 3 where axial diffusion is ignored. The computation from reference [ 121based on
0 I
0
I
2
DIMENSIONLESS
3
4
AXIAL DISTANCE,
5
6
7
X=Dx/R*uo
Fig. 4. Area-mean concentration distributions from present solution, plug flow model and one-dimensional model for p =@049, 0.071 and 0.092 with Pe = 100. -, present solution, Eq. (28); ---two-dimensional plug flow model, Eq. (29); ---one-dimensional model with coefficients given by Eqs. (A.24).(A.27)and(A.28).
0
02
04
DIMENSIONLESS
06
06
IO
RADIAL COORDINATE.
y:r/R
Fig. 3. Radial distributions of dimensionless local concentration at dimensionless axial distances X = 0.03 and 0.1 with /3 as -x=0.1
parameter and axial diffusion neglected; ___x
= o.03
thogonal expansion solution, Eq. (24).
Galerkin’s method and that from reference [9] based on a Runge-Kutta numerical integration of the convective diffusion equation are much more involved than the present one; comparison among these results, however, shows excellent agreement. It is clear from the figure that large rate constants and small diffusion coefficients can cause large transverse concentration gradients. Also note
As /? becomes larger, such as in liquid phase heterogeneous reactions, the use of the approximate models in kinetic interpretations of data could lead to erroneous results. For example, Fig. 5 shows that for /3 = 100the plug flow model can be in error by more than 50 per cent and the new one-dimensional model by about 50 per cent. The Walker model and the one-dimensional model based on Taylor’s dispersion theory give orders of magnitude deviations from the exact solution. The fraction of the reactant unconverted is plotted versus the dimensionless distance X for intermediate values of /3,say /3 = 1, in Fig. 6. Walker’s model, Eq. (30), and the one-dimensional model, Eq. (31), based on Taylor’s dispersion theory begin to deviate appreciably
690
W. N.
001
\j 0
004
0.08
0 12
016
GILL
I
0.20
024
0.28
DIMENSIONLESS AXIAL DISTANCE, X= Dx/R'uo
Fig. 5. Area-mean concentration distributions from present solution, plug flow model and one-dimensional models for fl = 100 and Pe = 100.1, one-dimensional model with coefficients given by Eq. (30); 2, one-dimensional model with coefficients given by Eq. (31); 3, two-dimensional plug flow model, Eq. (29); 4, generalized dispersion solution, Eq. (44); 5, present solution, Eq. (28); 6, one-dimensional model with coefficients given by Eqs. (A.24), (A.27)and (A.28). p=I
et al.
from the exact result, Eq. (28), for X >O.l. The new one-dimensional model, Eqs. (A.24), (A.27) and (A.28), and the plug flow model, Eq. (29), both give reasonable results with an error on the order of 10 per cent. Therefore one can conclude that while Walker’s model and Taylor’s model are applicable for p 5 0.1, the error becomes very large (orders of magnitude) for large p’s. The new model gives accurate results up to p = 1 and an error of only 50 per cent for large values of /I. (e) Generalized dispersion solution. Since neither the new model nor the plug flow model can predict the area-mean concentration distributions in a tubular reactor accurately for large values of p, a better approximation is necessary. In the previous approximation, the coefficients in the one-dimensional model were replaced by their asymptotic behavior for large times for the case where a pulse is introduced into the system. A better approximation for the case of constant concentration at X = 0 can be obtained by superposing the effect of an infinite number of pulses. The local concentration of the reactant before the steady state condition is achieved is described by the following dimensionless form of the convective diffusion equation: ~~+(]_#,lap+lgc
y ay
aX
ay Pe ax'
(33)
with appropriate conditions: C(0, X, YI= 0
(34a)
a?
(34b)
0, Y) = co
Pe=100
(34c) (34d) The basic methodology for solving Eqs. (33) and (34) without reaction at the wall has been developed by Sankarasubramanian and Gill[201. The presence of the first order chemical reaction at the wall does not affect the validity of the method. We shall give only the essential steps and the reader is referred to the above work for details. The problem represented by Eqs. (33) and (34) can be reformulated as: \ + \ 0.7
C(0, X Y) = 0
(364
DIMENSIONLESS AXIAL DISTANCE, X=Dr/R*",
Fig. 6. Area-mean concentration distributions from present solution, plug flow model and one-dimensional models for /3 = 1 and Pe = 100; 1, one-dimensional model with coefficients given by Eq. (30); 2, one-dimensional model with coefficients given by Eq. (31); 3, two-dimensional plug flow model, Eq. (29); 4, generalized dispersion solution, Eq. (44); 5, present solution, Eq. (28); 6, one-dimensional model with coefficients given by Eqs. (A.24),
(A.27)and (A.28).
C(7,X=-O,y)=O
(36b)
$T,x, 0) = 0
+,x,1) = - PC(T, X 1).
(36d)
691
Homogeneous models for porous catalysts and tubular reactors with heterogeneous reactions
The solution to the above problem can be obtained by superposing those solutions in response to a traln of pulses released at infinitely short time intervals. It is given as C(T, X, Y) = j07 CAT’, x, y) dr’
(37)
where C, satisfies the equation
$+(,_y’)aCr,ray~+L$G ax yay
ay
Pe
ax
(38)
(43)
where a = d/( I - (~KoKz/K~~)). This result is expected to be valid for relatively large T (or X). Since our concern here is the steady state distributions, the large time approximation should not affect the accuracy of the solution significantly. The steady state area-mean concentration distribution, determined from Eq. (43), is
along with
K, t $Kt
C,(O,Xl Y) = CoS(x)(l-
Y’)
$(T,x, 0) = 0 2(T,x, 1) = -
/3C,(T, x 1).
(39a) (39b) (39c)
Equations (38) and (39) describe the spreading of a non-uniform pulse in the presence of a first-order chemical reaction at the wall. This pulse problem has been analyzed [63and the area-mean concentration, C,,,,,can be shown to be
c,=4Peq(1$)~~~ Il-q Xl21 CO
(40)
where C(T)= I’Ko(s) ds
(41a)
Wb)
((7) =
1.‘Kds) ds.
(4lc)
The transient coefficients K,(T) and K*(T) are difficult to obtain due to the strong coupling in the system of associated equations as mentioned in the Appendix. It is therefore desirable to make certain approximations to the integrals in Eqs. (41). For large 7 (T L 0.5), the functions l(T), X~(T, X) and ((7) can be approximated by c(T)
x,(7,
-
x)-x+
VW
&T KIT
[(T) - KZT
Wb) (42~)
where KO,K, and K2 are the steady state values of Ko(T), K,(T) and K?(T) given in Eqs. (A.24), (A.27) and (A.28) of
the Appendix. Using the above approximations for Cl,,, and applying the superposition integral in Eq. (37) one obtains the following expression for the area-mean concentration for the case of constant inlet concentration:
Clll -=
CO
- 4KoK,lx
2K2 2V/(K,’ - 4KoKz)
I ’
(44
It is noted that both K. and K, are negative. The area-mean concentration is represented in Figs. 5 and 6. Figures 5 and 6 show that the generalized dispersion solution given by Eq. (44) agrees very well with the exact solution after X = 0.01 for /3 = 100 and after X i=0.1 for /3 = 1. These figures also show that the plug flow model provides a good approximation for small X Thus, the use of the generalized dispersion solution for relatively large X, along with the two-dimensional plug flow mode1 for small X, enables us to predict the area mean concentration over the entire range. Conclusions The improved one-dimensional
Thiele type model presented here for a cylindrical catalyst pore with a first-order reaction at the wall agrees well with the exact analysis of Bischoff. The improved model introduces an apparent volumetric reaction rate constant k: which depends on the diffusion coefficient as well as on the rate constant k, and the pore radius R. For a fast reaction with liquid reactant diffusing in a large pore, this new constant can differ from the traditional one by several orders of magnitude. As a result of the dependence of k: on the diffusion coefficient, the effectiveness factor is not uniquely determined by the Thiele modulus as in the traditional one-dimensional Thiele model. It also depends on the ratio, L, of the length of the pore to its radius. For many systems this ratio lies between 50 and 10’. For gaseous reactants and values of L in this range the Thiele’s mode1 provides a good approximation. For liquid reactants, however, the improved model presented here or Bischaff’s exact two-dimensional description should be used. A similar approach has been used to study a first order reaction catalyzed by the wall of a laminar flow tubular reactor. First, an exact orthogonal expansion solution to the problem was developed in terms of confluent hypergeometric functions. Compared to the existing exact solutions the new one has the advantage of accuracy and ease in computing the local as well as the area-mean concentrations. Second, a new one-dimensional mode1 is developed for tubular reactors. It provides a good approximation to the area-mean concentration for p up to 1.0, which covers all gas phase and some liquid phase
692
W. N. GILL et al.
heterogeneous reactions. For larger values of p, say p = lo*, the discrepancies between the exact solution and the present one-dimensional model are not greater than 50 per cent. The previous one-dimensional model of Walker[8] and that based on Taylor’s dispersion theory are accurate only for /3 5 0.1. They are useful only for gas phase heterogeneous reactions and give discrepancies of orders of magnitude for large values of /3 characteristic of liquid phase systems. To obtain more accurate predictions for p > 1, a generalized dispersion solution has been developed which gives a very good approximation after a short distance from thereactorentrance.Thiscombinedwithatwo-dimensional plug flow model, which is quite accurate in the upstream region, enables one to predict reactor performance satisfactorily for the entire reactor length.
NOTATION
expansion coefficients given in Eq. (A.22) expansion coefficients given in Eq. (24) coefficients given in Eq. (A.33) local concentration, ML-’ area-mean concentration = 2JfCrdr/R*, ML-’ local concentration for a pulse problem, MLm3 area-mean concentration = 2 _ffCrdr/R*, ML-’ inlet concentration, ML-’ molecular diffusion coefficient, L*T-’ functions defined in Eq. (4) integrals given in Eq. (A.32) Bessel functions of the first kind of zeroth order Jdx) Bessel functions of the first kind of first order ks reaction rate constant per unit surface, LT-’ k, traditional volumetric reaction rate constant, T-’
k: new volumetric reaction rate constant, T-’ k, coefficient defined in Eq. (3), LT-’ k2 coefficient defined in Eq. (3), L*T-’ Ki coefficients defined in Eq. (5) 1 half length of the catalyst pore, L L dimensionless half length of the catalyst pore ._ . M(a, b, z)confluent hypergeometric functtons Pe Peclet number = uoR/D r radial coordinate, L R radius, L t time, T u axial velocity in the tube, LT-’ UO axial velocity at the centerline of the tube, LT-’
dimensionless axial velocity = u /uO axial coordinate, L ; dimensionless axial coordinate, = x/R in catalyst problem, = Dx/R2uo in reactor problem function given in Eq. (41b) X1(7,X) dimensionless radial coordinate = r/R function defined in Eq. (A.2a) Y& Y” functions given in Eq. (25d)
dimensionless surface reaction rate constant = ksRID
Dirac delta function Kronecker delta defined in Eq. (A.7) function defined in Eq. (41a) effectiveness factor given in Eq. (1Oa) dimensionless local concentration for a pulse problem = G/G dimensionless area mean concentration = 2s: erydy eigenvalues given in Eq. (26) roots of Eq. (12) function defined in Eq. (41~) dimensionless time = Dt/R* Thiele modulus defined in Eq. (lla) function defined in Eq. (A.2a)
REFERENCES
[II Thiele E. W., Ind. Engng Cfiem. 1939 31 916. Dl SatterfieldC. N., Mass Transfer in Heterogeneous Catalysis, MIT Press, Cambridge, Mass., 1970. 131Thomas J. M. and Thomas W. J., Introduction to the Principles of HeterogeneousCatalysis,. Academic Press, New York, 1967. [41 Wheeler A., Adv. Catal. 1951III 249. PI Bischoff K. B., Ind. Engng Chem. Fund. 19665 135. bl Sankarasubramanian R. and Gill W. N., Proc. Roy. Sot. (Land.) 1973333A 115. [71 Baron T., Manning W. R. and Johnstone H. F., Chem. Engng Prog. 195248 125. 181Walker R. E., Phys. Fluids 19614 1211. 191Hsu C. J., Chem. Engng Sci. 196823 457. [lo] Katz S., Chem. Engng Sci. 195910 202. [l l] Krongelb S. and Strandberg M. W. P., J. Chem. Phys. 195931 11%. [12] Solomon R. L. and Hudson J. L., A.I.Ch.E. .U. 1%7 13545. 1131 _ - Gill W. N. and Sankarasubramanian R., Proc. Roy. Sot. (Land.) 1970316A 341. [14] Aris R., Proc. Roy. Sot. (Land.) 1956235A 67. [IS] Davis E. J., Can. .I Chem. Engng 197351 562. [16] Abramowitz M. and Stegun J. A. (Ed.), Handbook of Mathematical Functions, Dover, New York, 1968. [17] Hsu C. J., Appt. Sci. Res. 196717 359. [18] Damkijhler G. Z., Electrochem. 1936 42 846. 1191 Taylor G. I., Proc. Roy. Sot. (Land.) 1953219A 186. [20] Sankarasubramanian R. and Gill W. N., Proc. Roy. Sot. (Land.) 19723296 479. [21] Bischoff K. B., Ind. Engng Chem. Fund. 19665 285.
APPENDIX
When a pulse or a slug of solute is introduced into a tube at time t = 0, the dimensionless local concentration, 0, = C/C,, can be described by the following convective diffusion equation:
u
Greek
symbols
(Y constant given in Eq. (43)
along with the appropriate conditions &(0,X
Y)'wlY(Y)
%(T, x, 1) = - @,(T, x, 1)
(A.24
(A.2c)
e,(T,m,y) =
$(T,m, y) = 0.
(A.2d)
$,
0) = 0 (k = 0, 1,2, . . .).
The reaction catalyzed at the wall is assumed to be irreversible and first-order. The solution of Eqs. (A.1) and (A.2) is sought in the form:
$(?I)
(A.13
= - PMT, 1)
The area-mean concentration has to satisfy the compatibilit! conditions (A.14
where the dimensionless area-mean concentration I%,,,is defined by er, = 2 ’ e,y dy. I0
In addition, the definition of 8,,,,requires that
(A.4)
If Eq. (A.l) is multiplied by 2y and integrated from y = 0 to y = 1, the following dispersion equation for &Jr,,, is obtained after Eqs. (A.3) and (A.4) are taken into account:
I
O’h(~,y)ydy=~&,,
K~(T)= 2$(7,1) ay
(A.16
and
The coefficients Kg are given by
afo 1 a _=__ a7 y ay
Kh)=j$+2~(7,1)-2 I ‘fi-,(~,y)W,y)ydy 0
(i = O1lY2, ’ ’ ‘)(A6) where f-, =O and & is the Kronecker delta defined by
=0,
(A.15
where Sk0is the Kronecker delta defined by Eq. (A.7). For the functions f and K. one obtains from Eqs. (A.6) am (A.9)
(A.9
&,=I,
@=0,1,2,...)
afo
yT-foKo.
(A.17
The initial and boundary conditions on f. are given by Eqs. (A.12 and (A.13). Further, from Eq. (A.lS),
I,,’fey
i=J
i#j I ’
dy = ;.
(A.18
The final results for fO(T,y) and K0(7)can be shown to be [6] It has been shown[l3] that Eq. (A.5) can be truncated after the term involving K, without introducing serious errors. The resulting model for the area-mean concentration becomes
9 =&(T)e,,,, t K,(r)%+
Kd$.
U(r, y)fk-,+&ft-I-
:=”
(A.19
2 “zO(A./cL.)UP~) exp I - ~L.271
(A.@
The coefficients K, are related to the functions fk via equations which can be obtained by substituting Eq. (A.3) into Eq. (A.l) and using Eq. (AS) to calculate a&,,,/a7 and a *%, /aTax” in terms of a’&,,,/aX’. Equating the coefficients of a%,/aX’, (i = 0,1,2, . . .), one gets the following set of equations for fk:
$=+&y$-
x A.J&.Y 1w { - CLZTI
fo(T,Y)=
j. Kfu -‘o (k = 0, 1,2,. . J(A.9)
and
(A.20
where CL.are the roots of the transcendental equation
I
where f-, = 0 and fm2 = 0. The initial and boundary conditions on B,,,, and fk may be obtained from Eqs. (A.2) to (A.4) as follows: Since erm(o,x)=2
‘ye,(o,x,y)dy=2+(x)j’ I0
0
/L&L”) = &f&L.), (n = 0, 192,.. .I.
(A.21
The expansion coefficients A. are given by
A,,=
Y(Y)Ydy, (A.lO)
I’
Pn2 YY(Y)J&Y)~Y
’ (n = 0, 1,2,. . .). b.‘+ B’)L’(P.) 1’ YW) dy’ (A.22
fk(O,y)=O, (k=1,2,3
(A.ll)
,...)
As r+m, Eqs. (A.21) and (A.22) give the following asymptotil representations of f. and K.:
gives
f& Y)= &J~(P~Y) f (o y) = -= h(O, x, Y)
0 ,
edo, x)
2
Y(Y) 1 1n Y(Y)Ydy _-
.
(A.12)
Applying Eq. (A.3) to Eqs. (A.2b) and (A.2c) one obtains the following~boundaryconditions for fk are obtained:
CES Vol. 30, No. 7-D
(A.23
and K, =
Kc+)= - po
(A.24
where cl0 is the first (lowest in magnitude) root of Eq. (A.21).
694
W. N.
GILL et al.
Due to the strong coupling between ~JT, y) and G(T), the higher order functions are extremely diicult to calculate. Therefore we will content ourselves with the asymptotic steady-state representations of f*(~, y) and K*(T) for the case of steady flow. That is,
The integrals I(j, I) and the coefficients & are given by I(j I) = I(,
’
’
j)
=
2(282
= (“’ u(T,
)‘)
=
u(y)
=
+
cl:
+
Pz)
ccl:- cL:s
J&~V&~), ti+ 0 (A.31a)
p)~p~*(p’-3)J~(p,),
(j
= 1)
(A.J]b)
(A.25)
1-y’.
and When steady state is achieved, the f* will satisfy ;$y$+:jk
=(I -
y’)fk-,-&$-,+,$
f&X-,
(k = 1,2,3,. . J(A.26) The boundary conditions, Eqs. (A.13) and (AIS), still apply except that the r-dependence disappears. Thefinalresultsforthefk(y)andKk(k = 1,2,3,...)canbeshown to be 16) ~t/3’jPo’+(l-2BjPo2)2 2 6(l+ fi’/j~b)
1
(k=1,2,3 ,...;
j=l,2,3
,... ).
(A.32)
In obtaining Bj., the following conventions have been used: B,.-,=O
(A.27)
(j=O,1,2,3
B,.o=O Q=l,2,3
,...)
(A.33a)
,...)
(A.33b)
B,, = Jh2J4h) Bo., = -&,z k-l
- x K&,-r 1-3
1
, (k = 3,4,5,.
.)
(A.29)
(k = l,2,3,.
(k = 1,273,. . .h
(A.30)
.)
(A.33d)
and 21.1:&.J(L 0) BU = (P: - Po’)(B’+ /kVoI(P, )’
and X(y) = ,$ &.L(P,Y),
B,.+$,
(A.33~)
(A.33e)
It should be pointed out that the asymptotic values of the Kk and X(y) are independent of the initial solute distribution.
Pergamon Press.
Printed in Great Britain
HOMOGENEOUS MODELS FOR POROUS CATALYSTS AND TUBULAR REACTORS WITH HETEROGENEOUS REACTIONS W. N. GILL, E. RUCKENSTEIN? and H. P. HSIEH Department of Chemical Engineering, State University of New York at Buffalo, Buffalo,NY 14214,U.S.A. (Receioed 5 September 1974;nccepted 6 Nooember 1974)
Abstractane-dimensional models for porous catalysts and laminar flow tubular reactors with first-order reactions at the walls are developed. The method which is used replaces the transport equations containing the local concentrations with equations, similar to those used in the dispersion theory, for the concentrations averaged over the cross section The main assumption is the approximation of the coefficients in the dispersion equations by those valid for a pulse of concentration introduced at time zero at the mouth of the pore or at the inlet of the tube. The effectiveness factor depends not only on the Thiele modulus but also on the length-to-radius ratio of the catalyst pore. The one-dimensional mode! for the catalytic tubular reactor provides a good approximation to the area-mean concentration for p (= k,RID) up to 1.0. The comparison is made with an exact orthogonal expansion solution developed also in the paper. For larger values of /3,say ,9 = 100,the discrepancies between the exact solution and the one-dimensional model may be as high as 50 per cent. Therefore, a generalized dispersion solution is presented to obtain more accurate predictions for fi > I .O. INTRODUCTION
After Tbiele [l] an approximate one-dimensional
model
d2C D;i;r=kC has been used widely[24] to describe diffusion of a reacting species in a cylindrical pore with a first order catalytic reaction at the wall. The “volumetric” rate constant, $, is related to the intrinsic rate constant per unit surface area, ks, by
$2
R
where R is the radius of the pore. In Eq. (1) one assumes that the concentrations at the wall and in the bulk are essentially the same. Bischoff [5] analyzed the same problem accounting, however, for the transverse diffusion. The heterogeneous reaction was treated as a boundary condition. One of the objectives of this paper is to propose a model having the simplicity of Thiele’s model, but which takes into account the concentration variation over the cross-section of the pore. The concentration averaged over the cross-section is the main quantity used in the equation. The method by which the new model is constructed is borrowed from the theory of dispersion processes[6]. The new model, which will be shown to produce results which agree well with BischotI’s results, indicates that the volumetric rate constant depends not only on k, and R as suggested by Eq. (2), but also on the pore diffusion coefficient D. The second goal of the paper is to apply the same procedure to more complex problems, for instance, to tTo whom all the correspondence should be addressed.
laminar flow in a tubular reactor with a wall-catalyzed reaction of the first order. The tubular flow reactor with catalyst deposited on the wall is useful for obtaining precise kinetic data from conversion measurements in view of its well-defined flow field[7]. Walker[8] suggested a one-dimensional model to describe the area-mean concentration in the tubular reactor: d2C,
dC,
kzdX2 + kldx
= kucm
where k, is given by Eq. (2), k, is equal to the negative of the average velocity and k2 is the molecular diffusion coefficient. Our goal is to develop an equation for the area-mean concentration of the same form, but without the assumption that the wall and the bulk concentrations are equal. The results obtained by this procedure will be compared with Walker’s result and with an exact solution. Although exact solutions are available[%12], a new solution is derived which has the advantage of computational ease and higher accuracy. The method which will be used to develop the new one-dimensional models for both the porous catalyst and the tubular reactor replaces the transport equation containing the local concentration with an equation for the concentration averaged over the cross-section. The equation is similar to those used in dispersion problems. The main assumption consists of approximating the coefficients in the dispersion equation by those valid for a pulse of concentration introduced at the initial moment at the mouth of the pore or at the inlet of the tube. The computation contains the following steps: (a) With a pulse (or a slug) introduced into a tube or a pore, the dispersion theory of Gill and Sankarasubramanian [6] expands the local concentration in
686
W. N. GILLetal.
series of the form (4) where Cr,,, is the area-mean concentration, X is the dimensionless axial coordinate and the functions X depend on the radial coordinate. (b) The area-mean concentration is formulated to follow the generalized dispersion equation
where r is a dimensionless time. It has been shown]131 that Eq. (5) can be truncated after the term involving KZ without introducing serious errors. The functions h and the coefficients Ki are related to each other and can be determined as shown in the Appendix. (c) The steady state values of K,, K, and Kt obtained for a pulse (or a slug) problem then are assumed to be directly applicable to steady state dispersion problems with constant inlet concentrations. That is, the coefficients K,,, K, and Kz, valid for a pulse, are used in the following equation to describe the steady state area-mean concentration in a pore or a tube: d?,,, dC,,, Kz+ K,z + KoC, = 0
Exact solution Bischoff [5] analyzed the above problem exactly and obtained for the local concentration
For the effectiveness factor, 7, he derived the equation:
(lot.4 . where 4 is the Thiele modulus defined by [3-51 (lla)
L = l/R.
(lib)
and
The PI’S are determined from the transcendental equation
(6)
with the constant inlet condition C(X = 0) = C,(X = 0) = c0. For large values of the catalytic parameter P(= k,R/D), Eq. (6) fails to describe tubular reactors with reactions on the wall accurately. A new result (Eq. 44) is given, which is relatively simple and accurate, for such systems.
Zk, “* eS=l m = LV(2B) ( >
pJdpi) = pI&i),
(7) The boundary conditions for a first-order chemical reaction at the wall are:
(12)
where ,3 ~ k RID s . NeW one_dimensional model The method outlined in the introduction gives, for systems without convection (see Appendix), KO=-pd
I.CYLINDRICALCATALYST PORES
Gooeming equations Consider a cylindrical pore of radius R and length 21 in a porous catalyst pellet. Neglecting any convection in the pore, the local concentration at steady state is described by
(i = 0, 1,2,3.. .I
K,=O
(13b)
K2= 1,
(13c)
and
where p. is the first (lowest in absolute magnitude) root of Eq. (12). The new model for cylindrical catalyst pores, therefore, becomes d2C, 3 = /.klV?tl
(14)
or, in terms of the dimensional coordinate x,
$(x,0) = 0
-+x,
R) = k,C(x, R)
(84 (W
d2C, _ D,x2-kbC,w The new volumetric rate constant, kd is given by
C(0, r) = Co
$1,r) = 0.
(15)
(16) (84
where p. depends on p.
Homogeneous models for porous catalysts and tubular reactors with heterogeneous reactions
The area-mean concentration, C,,,, has to satisfy the area-average forms of Eqs. (8~) and (8d). With the above developments, one obtains C L= co
cash [$‘(l -x)] cash (/&IR)
(17)
(b) Effectioeness factors. As a consequence of the dependence of k: on the diffusion coefficient, the new model predicts that the effectiveness factor can be significantly different from that obtained from the traditional model. The effectiveness factor computed from the traditional model is
and consequently the effectiveness factor is given by:
Discussion
(a) Volumetric rate constants. Comparison between Eqs. (1) and (15) indicates that a new volumetric rate constant k: should replace the traditional constant k,( = 2k,/R) in the one-dimensional models for cylindrical catalyst pores. The new constant depends on the diffusion coefficient D as well as on k, and R. The relation between these two constants has the form kl /LO -=k,
n=-
and is shown in Fig. I. As the wall catalytic parameter p approaches unity, the constant k. begins to deviate appreciably from the new constant kl (by nearly 30 per cent). For small values of p, Eq. (12) leads to pLo= v(2fi) and therefore kb- k,.
(2W
tanh 4 dJ ’
(21)
Both Bischoff’s result and the new model predict that the effectiveness factor depends not only on the Thiele modulus but also on the ratio L( = l/R). For large values of L Eq. (18) reduces to Eq. (21). The effectiveness factors obtained from the three approaches are given in Fig. 2. The differences between the results of the new model, Eq. (18), and those from Bischotf’s analysis, Eq. (lob), are fairly small over the range of practical interest. IO
(19)
2P
687
01
F 8 5 Y
0.01
8
Y ; B :: w
0.001
For large values of /3, pa = 2.4048 and ,+28916& P
“.
(2Ob)
These asymptotic results are included in Fig. 1. As expected, k, is close to k: when the reaction rate is slow. For instance, up to /? = 0.2, k, differs from k: by less than about 5 per cent. However, as /3 becomes larger, the ratio &L/k, becomes linear in l/p with a slope of 2.8916 and the two constants can differ by orders of magnitude. This happens, for example, when a relatively fast reaction occurs at the wall of a large pore in which a liquid reactant is diffusing.
00001
0.10
IO
Kl THIELE
MODULUS.
100
1000
#I
Fig. 2. Effectiveness factor vs Thiele modulus with dimensionless pore length as parameter. -, improved one-dimensional model, Eq. (18);---, Bischoff’s result, Eq. (lob).
However, the new model involves only one term and therefore is simpler to use than Bischotf’s expression. The figure shows that the traditional model may overestimate the effectiveness factor by orders of magnitude, especially when the pores are short in length and large in radius. This overestimation becomes serious when the Thiele modulus is large. Equation (1la) shows that, for a given finite pore length, /3 increases as the Thiele modulus 4 increases. When p becomes sufficiently huge, p. approaches 2.4048 and Eq. (18)givestheasymptoticresultconjectured byAris[21] ~ = (24048L) tanh (2.4048L) _ 1 6* P
(21a)
By contrast, the traditional model leads to
9--
Fig. 1. Ratio of the new to traditional volumetric rate constants vs the catalytic parameter 8.
for large p.
1
@lb)
W. N. GILL et al.
688
For many catalysts the length-to-radius ratios lie between 50 and 10’[5]. Therefore for gaseous reactants the effectiveness factor is close to the traditional predictions because they have small or moderate Thiele moduli. For liquid reactants, because of the low diffusion coefficient, the Thiele modulus is large, and therefore the new model is more appropriate than the traditional one.
IL WALL-CATALYZED TURULAR REACTORS
Describing equations
The procedure used in the previous section is applied to the case of laminar flow in a tubular reactor with first-order reactions at the wall. The reactant concentration in the tube of radius R is described by
are given by y”(y)
I A y2 e-*PY2M 2 - A.(1 + A.2/Pe2),,n ,
(
=
4
) (n = 0, 1,2,. . .)(25d)
where M(a, b, z) is the confluent hypergeometric function tabulated, for instance, in reference[l6]. Substituting Eq. (25d) into Eq. (25~) gives the following transcendental equation for the A.: 2A. - A:(1 + Az/Pe2)M 6- A,(1 + A~IPe’) 2 A , , n 4 2 2 - A.(I; A.‘/Pe’), 1, A. = o, > (n = 0, 1,2,. . J. (26)
+(p -A.)M
(
The coefficients B. from Eq. (24) can be calculated on the basis of the orthogonal properties of Y, ( y ) [9,17]. The final result is where u. is the centerline velocity. The concentration at the inlet is assumed to be constant, therefore, C(0, r) = CO.
(234
To complete the description of the system, the following boundary conditions are written:
With the local concentration given by Eq. (24) one obtains the area-mean concentration as m
$(x, 0) = 0
-+x,
R) = k,C(x, R)
~=2~~E”e-*:Xf’yY,(y)dy. 0
(23b)
(23~)
where k, is the heterogeneous rate constant. (a) Exact solution. Although exact solutions for Eqs. (22) and (23) are available[%12] another solution will be developed. Results are more accurate and easier to compute from this solution. The solution is formulated in terms of dimensionless axial and radial coordinates X and y as
(28)
(b) Two-dimensional plug flow model. Damkiihler [ 181 and Baron et al. [7] simplified .the problem by assuming a rod-like flow pattern with the velocity everywhere replaced by the average velocity of the flow. Their result accordingly becomes
oI exp{-y(J(lt$)-l)] %=4c
n=O
(29)
p1[1+(;)‘]
where the p.‘s are calculated from Eq. (12). -_ cs-2
B,e-A2XY.(y)
(24)
where A,,and Y.(y) can be shown to satisfy the relations ;gyF+Az
[
(Fy’)+$1Y" =o
$0) =0
W-4 (W
and
gql, =- pY"(l).
(25~)
It has been shownU51 that Y,, satisfying Eqs. (25aH25c)
(c) Previous one-dimensional models. Incontrast to the two-dimensional approaches described above the one dimensional models of the form of Eq. (6) for the area-mean concentration are simpler and hence easier to use. Walker[8] suggested a one-dimensional model with the coefficients K. = - 2p;
K, = - l/2;
KZ = l/Pe’.
(30)
Equation (6) with the coefficients K1 and K2 taken from the Taylor&is’ dispersion theory without reaction[l4,19] has been widely used in modeling homogeneous reactions in tubular reactors. Applying the same concept to heterogeneous reactions one obtains: K,,=-2p;
K,=-l/2;
Kz=1/192+1/Pe2.
(31)
689
Homogeneous models for porous catalysts and tubular reactors with heterogeneous reactions
(d) New one-dimensional model. As mentioned earlier, the coefficients I&, K, and X2 from Eqs. (A.24), (A.27) and (A.28) in the Appendix are valid for a pulse (or a slug) introduced into a pore or a tube with first order reactions at the wall. They are used to approximate the corresponding coefficients in the one-dimensional model for a step change at the inlet. Note that the coefficients in the new model differ from those given by Eq. (31) since the former takes into account the heterogeneous reactions at the wall. The concentration distribution from the onedimensional models, Eq. (6), can be shown to be Clll
Co=exp
_ K, + V(KIZ - 4&&)X 2K2 1.
(32)
It should be noted that the coefficient & is negative because reactant is consumed at the wall by the catalytic reaction.
that the centerline concentration at X =0*03 remains approximately the same for all p’s, while that at X = O-1 decreases appreciably as /3 increases. This is mainly due to dispersion, which is the combined action of axial convection and radial diffusion. For gas phase wall-catalyzed reactions the value of the catalytic parameter /3 is less than about 0.1[7,12]. The area-mean concentration distributions obtained from the new one-dimensional model, Eqs. (A.24), (A.27) and (A.28), are found to deviate from the exact solution by less than 3 per cent for the practical range of distances. For /3 SO.1 the new model gives results similar to those obtained by other models. For clarity of the presentation, in Fig. 4 only the new one-dimensional model and the two-dimensional plug flow model are compared with the exact solution.
Discussion Radial concentration profiles obtained from the exact solution, Eq. (24), are shown in Fig. 3 where axial diffusion is ignored. The computation from reference [ 121based on
0 I
0
I
2
DIMENSIONLESS
3
4
AXIAL DISTANCE,
5
6
7
X=Dx/R*uo
Fig. 4. Area-mean concentration distributions from present solution, plug flow model and one-dimensional model for p =@049, 0.071 and 0.092 with Pe = 100. -, present solution, Eq. (28); ---two-dimensional plug flow model, Eq. (29); ---one-dimensional model with coefficients given by Eqs. (A.24).(A.27)and(A.28).
0
02
04
DIMENSIONLESS
06
06
IO
RADIAL COORDINATE.
y:r/R
Fig. 3. Radial distributions of dimensionless local concentration at dimensionless axial distances X = 0.03 and 0.1 with /3 as -x=0.1
parameter and axial diffusion neglected; ___x
= o.03
thogonal expansion solution, Eq. (24).
Galerkin’s method and that from reference [9] based on a Runge-Kutta numerical integration of the convective diffusion equation are much more involved than the present one; comparison among these results, however, shows excellent agreement. It is clear from the figure that large rate constants and small diffusion coefficients can cause large transverse concentration gradients. Also note
As /? becomes larger, such as in liquid phase heterogeneous reactions, the use of the approximate models in kinetic interpretations of data could lead to erroneous results. For example, Fig. 5 shows that for /3 = 100the plug flow model can be in error by more than 50 per cent and the new one-dimensional model by about 50 per cent. The Walker model and the one-dimensional model based on Taylor’s dispersion theory give orders of magnitude deviations from the exact solution. The fraction of the reactant unconverted is plotted versus the dimensionless distance X for intermediate values of /3,say /3 = 1, in Fig. 6. Walker’s model, Eq. (30), and the one-dimensional model, Eq. (31), based on Taylor’s dispersion theory begin to deviate appreciably
690
W. N.
001
\j 0
004
0.08
0 12
016
GILL
I
0.20
024
0.28
DIMENSIONLESS AXIAL DISTANCE, X= Dx/R'uo
Fig. 5. Area-mean concentration distributions from present solution, plug flow model and one-dimensional models for fl = 100 and Pe = 100.1, one-dimensional model with coefficients given by Eq. (30); 2, one-dimensional model with coefficients given by Eq. (31); 3, two-dimensional plug flow model, Eq. (29); 4, generalized dispersion solution, Eq. (44); 5, present solution, Eq. (28); 6, one-dimensional model with coefficients given by Eqs. (A.24), (A.27)and (A.28). p=I
et al.
from the exact result, Eq. (28), for X >O.l. The new one-dimensional model, Eqs. (A.24), (A.27) and (A.28), and the plug flow model, Eq. (29), both give reasonable results with an error on the order of 10 per cent. Therefore one can conclude that while Walker’s model and Taylor’s model are applicable for p 5 0.1, the error becomes very large (orders of magnitude) for large p’s. The new model gives accurate results up to p = 1 and an error of only 50 per cent for large values of /I. (e) Generalized dispersion solution. Since neither the new model nor the plug flow model can predict the area-mean concentration distributions in a tubular reactor accurately for large values of p, a better approximation is necessary. In the previous approximation, the coefficients in the one-dimensional model were replaced by their asymptotic behavior for large times for the case where a pulse is introduced into the system. A better approximation for the case of constant concentration at X = 0 can be obtained by superposing the effect of an infinite number of pulses. The local concentration of the reactant before the steady state condition is achieved is described by the following dimensionless form of the convective diffusion equation: ~~+(]_#,lap+lgc
y ay
aX
ay Pe ax'
(33)
with appropriate conditions: C(0, X, YI= 0
(34a)
a?
(34b)
0, Y) = co
Pe=100
(34c) (34d) The basic methodology for solving Eqs. (33) and (34) without reaction at the wall has been developed by Sankarasubramanian and Gill[201. The presence of the first order chemical reaction at the wall does not affect the validity of the method. We shall give only the essential steps and the reader is referred to the above work for details. The problem represented by Eqs. (33) and (34) can be reformulated as: \ + \ 0.7
C(0, X Y) = 0
(364
DIMENSIONLESS AXIAL DISTANCE, X=Dr/R*",
Fig. 6. Area-mean concentration distributions from present solution, plug flow model and one-dimensional models for /3 = 1 and Pe = 100; 1, one-dimensional model with coefficients given by Eq. (30); 2, one-dimensional model with coefficients given by Eq. (31); 3, two-dimensional plug flow model, Eq. (29); 4, generalized dispersion solution, Eq. (44); 5, present solution, Eq. (28); 6, one-dimensional model with coefficients given by Eqs. (A.24),
(A.27)and (A.28).
C(7,X=-O,y)=O
(36b)
$T,x, 0) = 0
+,x,1) = - PC(T, X 1).
(36d)
691
Homogeneous models for porous catalysts and tubular reactors with heterogeneous reactions
The solution to the above problem can be obtained by superposing those solutions in response to a traln of pulses released at infinitely short time intervals. It is given as C(T, X, Y) = j07 CAT’, x, y) dr’
(37)
where C, satisfies the equation
$+(,_y’)aCr,ray~+L$G ax yay
ay
Pe
ax
(38)
(43)
where a = d/( I - (~KoKz/K~~)). This result is expected to be valid for relatively large T (or X). Since our concern here is the steady state distributions, the large time approximation should not affect the accuracy of the solution significantly. The steady state area-mean concentration distribution, determined from Eq. (43), is
along with
K, t $Kt
C,(O,Xl Y) = CoS(x)(l-
Y’)
$(T,x, 0) = 0 2(T,x, 1) = -
/3C,(T, x 1).
(39a) (39b) (39c)
Equations (38) and (39) describe the spreading of a non-uniform pulse in the presence of a first-order chemical reaction at the wall. This pulse problem has been analyzed [63and the area-mean concentration, C,,,,,can be shown to be
c,=4Peq(1$)~~~ Il-q Xl21 CO
(40)
where C(T)= I’Ko(s) ds
(41a)
Wb)
((7) =
1.‘Kds) ds.
(4lc)
The transient coefficients K,(T) and K*(T) are difficult to obtain due to the strong coupling in the system of associated equations as mentioned in the Appendix. It is therefore desirable to make certain approximations to the integrals in Eqs. (41). For large 7 (T L 0.5), the functions l(T), X~(T, X) and ((7) can be approximated by c(T)
x,(7,
-
x)-x+
VW
&T KIT
[(T) - KZT
Wb) (42~)
where KO,K, and K2 are the steady state values of Ko(T), K,(T) and K?(T) given in Eqs. (A.24), (A.27) and (A.28) of
the Appendix. Using the above approximations for Cl,,, and applying the superposition integral in Eq. (37) one obtains the following expression for the area-mean concentration for the case of constant inlet concentration:
Clll -=
CO
- 4KoK,lx
2K2 2V/(K,’ - 4KoKz)
I ’
(44
It is noted that both K. and K, are negative. The area-mean concentration is represented in Figs. 5 and 6. Figures 5 and 6 show that the generalized dispersion solution given by Eq. (44) agrees very well with the exact solution after X = 0.01 for /3 = 100 and after X i=0.1 for /3 = 1. These figures also show that the plug flow model provides a good approximation for small X Thus, the use of the generalized dispersion solution for relatively large X, along with the two-dimensional plug flow mode1 for small X, enables us to predict the area mean concentration over the entire range. Conclusions The improved one-dimensional
Thiele type model presented here for a cylindrical catalyst pore with a first-order reaction at the wall agrees well with the exact analysis of Bischoff. The improved model introduces an apparent volumetric reaction rate constant k: which depends on the diffusion coefficient as well as on the rate constant k, and the pore radius R. For a fast reaction with liquid reactant diffusing in a large pore, this new constant can differ from the traditional one by several orders of magnitude. As a result of the dependence of k: on the diffusion coefficient, the effectiveness factor is not uniquely determined by the Thiele modulus as in the traditional one-dimensional Thiele model. It also depends on the ratio, L, of the length of the pore to its radius. For many systems this ratio lies between 50 and 10’. For gaseous reactants and values of L in this range the Thiele’s mode1 provides a good approximation. For liquid reactants, however, the improved model presented here or Bischaff’s exact two-dimensional description should be used. A similar approach has been used to study a first order reaction catalyzed by the wall of a laminar flow tubular reactor. First, an exact orthogonal expansion solution to the problem was developed in terms of confluent hypergeometric functions. Compared to the existing exact solutions the new one has the advantage of accuracy and ease in computing the local as well as the area-mean concentrations. Second, a new one-dimensional mode1 is developed for tubular reactors. It provides a good approximation to the area-mean concentration for p up to 1.0, which covers all gas phase and some liquid phase
692
W. N. GILL et al.
heterogeneous reactions. For larger values of p, say p = lo*, the discrepancies between the exact solution and the present one-dimensional model are not greater than 50 per cent. The previous one-dimensional model of Walker[8] and that based on Taylor’s dispersion theory are accurate only for /3 5 0.1. They are useful only for gas phase heterogeneous reactions and give discrepancies of orders of magnitude for large values of /3 characteristic of liquid phase systems. To obtain more accurate predictions for p > 1, a generalized dispersion solution has been developed which gives a very good approximation after a short distance from thereactorentrance.Thiscombinedwithatwo-dimensional plug flow model, which is quite accurate in the upstream region, enables one to predict reactor performance satisfactorily for the entire reactor length.
NOTATION
expansion coefficients given in Eq. (A.22) expansion coefficients given in Eq. (24) coefficients given in Eq. (A.33) local concentration, ML-’ area-mean concentration = 2JfCrdr/R*, ML-’ local concentration for a pulse problem, MLm3 area-mean concentration = 2 _ffCrdr/R*, ML-’ inlet concentration, ML-’ molecular diffusion coefficient, L*T-’ functions defined in Eq. (4) integrals given in Eq. (A.32) Bessel functions of the first kind of zeroth order Jdx) Bessel functions of the first kind of first order ks reaction rate constant per unit surface, LT-’ k, traditional volumetric reaction rate constant, T-’
k: new volumetric reaction rate constant, T-’ k, coefficient defined in Eq. (3), LT-’ k2 coefficient defined in Eq. (3), L*T-’ Ki coefficients defined in Eq. (5) 1 half length of the catalyst pore, L L dimensionless half length of the catalyst pore ._ . M(a, b, z)confluent hypergeometric functtons Pe Peclet number = uoR/D r radial coordinate, L R radius, L t time, T u axial velocity in the tube, LT-’ UO axial velocity at the centerline of the tube, LT-’
dimensionless axial velocity = u /uO axial coordinate, L ; dimensionless axial coordinate, = x/R in catalyst problem, = Dx/R2uo in reactor problem function given in Eq. (41b) X1(7,X) dimensionless radial coordinate = r/R function defined in Eq. (A.2a) Y& Y” functions given in Eq. (25d)
dimensionless surface reaction rate constant = ksRID
Dirac delta function Kronecker delta defined in Eq. (A.7) function defined in Eq. (41a) effectiveness factor given in Eq. (1Oa) dimensionless local concentration for a pulse problem = G/G dimensionless area mean concentration = 2s: erydy eigenvalues given in Eq. (26) roots of Eq. (12) function defined in Eq. (41~) dimensionless time = Dt/R* Thiele modulus defined in Eq. (lla) function defined in Eq. (A.2a)
REFERENCES
[II Thiele E. W., Ind. Engng Cfiem. 1939 31 916. Dl SatterfieldC. N., Mass Transfer in Heterogeneous Catalysis, MIT Press, Cambridge, Mass., 1970. 131Thomas J. M. and Thomas W. J., Introduction to the Principles of HeterogeneousCatalysis,. Academic Press, New York, 1967. [41 Wheeler A., Adv. Catal. 1951III 249. PI Bischoff K. B., Ind. Engng Chem. Fund. 19665 135. bl Sankarasubramanian R. and Gill W. N., Proc. Roy. Sot. (Land.) 1973333A 115. [71 Baron T., Manning W. R. and Johnstone H. F., Chem. Engng Prog. 195248 125. 181Walker R. E., Phys. Fluids 19614 1211. 191Hsu C. J., Chem. Engng Sci. 196823 457. [lo] Katz S., Chem. Engng Sci. 195910 202. [l l] Krongelb S. and Strandberg M. W. P., J. Chem. Phys. 195931 11%. [12] Solomon R. L. and Hudson J. L., A.I.Ch.E. .U. 1%7 13545. 1131 _ - Gill W. N. and Sankarasubramanian R., Proc. Roy. Sot. (Land.) 1970316A 341. [14] Aris R., Proc. Roy. Sot. (Land.) 1956235A 67. [IS] Davis E. J., Can. .I Chem. Engng 197351 562. [16] Abramowitz M. and Stegun J. A. (Ed.), Handbook of Mathematical Functions, Dover, New York, 1968. [17] Hsu C. J., Appt. Sci. Res. 196717 359. [18] Damkijhler G. Z., Electrochem. 1936 42 846. 1191 Taylor G. I., Proc. Roy. Sot. (Land.) 1953219A 186. [20] Sankarasubramanian R. and Gill W. N., Proc. Roy. Sot. (Land.) 19723296 479. [21] Bischoff K. B., Ind. Engng Chem. Fund. 19665 285.
APPENDIX
When a pulse or a slug of solute is introduced into a tube at time t = 0, the dimensionless local concentration, 0, = C/C,, can be described by the following convective diffusion equation:
u
Greek
symbols
(Y constant given in Eq. (43)
along with the appropriate conditions &(0,X
Y)'wlY(Y)
%(T, x, 1) = - @,(T, x, 1)
(A.24
(A.2c)
e,(T,m,y) =
$(T,m, y) = 0.
(A.2d)
$,
0) = 0 (k = 0, 1,2, . . .).
The reaction catalyzed at the wall is assumed to be irreversible and first-order. The solution of Eqs. (A.1) and (A.2) is sought in the form:
$(?I)
(A.13
= - PMT, 1)
The area-mean concentration has to satisfy the compatibilit! conditions (A.14
where the dimensionless area-mean concentration I%,,,is defined by er, = 2 ’ e,y dy. I0
In addition, the definition of 8,,,,requires that
(A.4)
If Eq. (A.l) is multiplied by 2y and integrated from y = 0 to y = 1, the following dispersion equation for &Jr,,, is obtained after Eqs. (A.3) and (A.4) are taken into account:
I
O’h(~,y)ydy=~&,,
K~(T)= 2$(7,1) ay
(A.16
and
The coefficients Kg are given by
afo 1 a _=__ a7 y ay
Kh)=j$+2~(7,1)-2 I ‘fi-,(~,y)W,y)ydy 0
(i = O1lY2, ’ ’ ‘)(A6) where f-, =O and & is the Kronecker delta defined by
=0,
(A.15
where Sk0is the Kronecker delta defined by Eq. (A.7). For the functions f and K. one obtains from Eqs. (A.6) am (A.9)
(A.9
&,=I,
@=0,1,2,...)
afo
yT-foKo.
(A.17
The initial and boundary conditions on f. are given by Eqs. (A.12 and (A.13). Further, from Eq. (A.lS),
I,,’fey
i=J
i#j I ’
dy = ;.
(A.18
The final results for fO(T,y) and K0(7)can be shown to be [6] It has been shown[l3] that Eq. (A.5) can be truncated after the term involving K, without introducing serious errors. The resulting model for the area-mean concentration becomes
9 =&(T)e,,,, t K,(r)%+
Kd$.
U(r, y)fk-,+&ft-I-
:=”
(A.19
2 “zO(A./cL.)UP~) exp I - ~L.271
(A.@
The coefficients K, are related to the functions fk via equations which can be obtained by substituting Eq. (A.3) into Eq. (A.l) and using Eq. (AS) to calculate a&,,,/a7 and a *%, /aTax” in terms of a’&,,,/aX’. Equating the coefficients of a%,/aX’, (i = 0,1,2, . . .), one gets the following set of equations for fk:
$=+&y$-
x A.J&.Y 1w { - CLZTI
fo(T,Y)=
j. Kfu -‘o (k = 0, 1,2,. . J(A.9)
and
(A.20
where CL.are the roots of the transcendental equation
I
where f-, = 0 and fm2 = 0. The initial and boundary conditions on B,,,, and fk may be obtained from Eqs. (A.2) to (A.4) as follows: Since erm(o,x)=2
‘ye,(o,x,y)dy=2+(x)j’ I0
0
/L&L”) = &f&L.), (n = 0, 192,.. .I.
(A.21
The expansion coefficients A. are given by
A,,=
Y(Y)Ydy, (A.lO)
I’
Pn2 YY(Y)J&Y)~Y
’ (n = 0, 1,2,. . .). b.‘+ B’)L’(P.) 1’ YW) dy’ (A.22
fk(O,y)=O, (k=1,2,3
(A.ll)
,...)
As r+m, Eqs. (A.21) and (A.22) give the following asymptotil representations of f. and K.:
gives
f& Y)= &J~(P~Y) f (o y) = -= h(O, x, Y)
0 ,
edo, x)
2
Y(Y) 1 1n Y(Y)Ydy _-
.
(A.12)
Applying Eq. (A.3) to Eqs. (A.2b) and (A.2c) one obtains the following~boundaryconditions for fk are obtained:
CES Vol. 30, No. 7-D
(A.23
and K, =
Kc+)= - po
(A.24
where cl0 is the first (lowest in magnitude) root of Eq. (A.21).
694
W. N.
GILL et al.
Due to the strong coupling between ~JT, y) and G(T), the higher order functions are extremely diicult to calculate. Therefore we will content ourselves with the asymptotic steady-state representations of f*(~, y) and K*(T) for the case of steady flow. That is,
The integrals I(j, I) and the coefficients & are given by I(j I) = I(,
’
’
j)
=
2(282
= (“’ u(T,
)‘)
=
u(y)
=
+
cl:
+
Pz)
ccl:- cL:s
J&~V&~), ti+ 0 (A.31a)
p)~p~*(p’-3)J~(p,),
(j
= 1)
(A.J]b)
(A.25)
1-y’.
and When steady state is achieved, the f* will satisfy ;$y$+:jk
=(I -
y’)fk-,-&$-,+,$
f&X-,
(k = 1,2,3,. . J(A.26) The boundary conditions, Eqs. (A.13) and (AIS), still apply except that the r-dependence disappears. Thefinalresultsforthefk(y)andKk(k = 1,2,3,...)canbeshown to be 16) ~t/3’jPo’+(l-2BjPo2)2 2 6(l+ fi’/j~b)
1
(k=1,2,3 ,...;
j=l,2,3
,... ).
(A.32)
In obtaining Bj., the following conventions have been used: B,.-,=O
(A.27)
(j=O,1,2,3
B,.o=O Q=l,2,3
,...)
(A.33a)
,...)
(A.33b)
B,, = Jh2J4h) Bo., = -&,z k-l
- x K&,-r 1-3
1
, (k = 3,4,5,.
.)
(A.29)
(k = l,2,3,.
(k = 1,273,. . .h
(A.30)
.)
(A.33d)
and 21.1:&.J(L 0) BU = (P: - Po’)(B’+ /kVoI(P, )’
and X(y) = ,$ &.L(P,Y),
B,.+$,
(A.33~)
(A.33e)
It should be pointed out that the asymptotic values of the Kk and X(y) are independent of the initial solute distribution.