Ind. Eng. Chem. Res. 1995,34, 3771-3776
3771
Rational Approximation of the Overall Effectiveness Factor for the Gas-Liquid-Solid Phase Catalytic Reaction Qiuxiang Yin and Shaofen Li* Department of Chemical Engineering, Tianjin University, Tianjin 300072, People’s Republic of China
The case of gas-liquid-solid phase catalytic reaction with external transfer resistances is analyzed. A rational approximate expression for the overall effectiveness factor, which is considered to be a function of the Thiele modulus and Damkohler number, is derived after matching asymptotic solutions valid for small and large values of Thiele moduli. As compared with the analytical and numerical solutions, the accuracy of the approximate expression is surprisingly high. I n addition, this method does not require any iterative or trial-and-error calculations, and should offer significant time savings in the simulations of heterogeneous catalytic reactors.
Introduction For the gas-liquid-solid phase catalytic reaction process, a number of steps such as gas-liquid mass transfer, liquid-solid mass transfer, intraparticle diffusion, and chemical reaction are involved for a species present in the gas phase to react a t the active sites of the catalyst. The overall effectiveness factor, defined as the ratio of the actual rate of reaction t o the rate which would occur in the absence of all the transfer resistances, is a useful concept in analyzing such complex processes. Since the reaction-diffusion equation for the catalyst pellet constitutes a nonlinear boundary value problem, an exact analytical expression for the overall effectiveness factor is impossible for a non-first-order reaction. Numerical methods which are always time-consuming must be employed. Therefore the need arises for an approximate procedure which allows a rapid calculation of the overall effectiveness factor with an acceptable accuracy, particularly in the simulations of fmed-bed reactors in which the overall effectiveness factor is affected by local conditions and has t o be determined repeatedly for all positions within the reactor. Many efforts (Wedel and Luss, 1980; Gottifredi et al., 1981a,b; Gonzo and Gottifredi, 1983; Yang and Li, 1988) have been made t o develop approximate expressions for the effectiveness factor based on matching techniques from the knowledge of asymptotic solutions. However, these approximate solutions are valid only in the absence of external transfer resistances. For the case of external resistances, as far as the authors can ascertain, there is still no analytical or approximate procedure to estimate rapidly the overall effectiveness factor for the whole range of Thiele moduli. Gottifredi et al. (1981a,b) suggested that the trial-and-error procedure could be used to compute the overall effectiveness factor based on the approximate expression valid in the absence of external resistances. Ramachandran and Chaudhari (1979, 1980) used the analytical solution valid for the first-order reaction to estimate the overall effectiveness factor for non-first-order reaction based on the concept of a generalized Thiele modulus (Bischoff, 1965) and an iteration procedure. Obviously, these methods have not yet achieved the goal to save time. The aim of this work is t o present a general method for computing rapidly an approximate value of the
* Author t o whom correspondence should be addressed.
overall effectiveness factor for the gas-liquid-solid phase catalytic reaction in which external mass transfer resistances are taken into account. We consider the overall effectiveness factor to be a function of the Thiele modulus and the Damkohler number, which account for the effects of the internal and external resistances, respectively, with other parameters entering as required by the specific rate expression.
Approximate Analysis A general gas-liquid-solid phase catalytic reaction system can be represented by the following scheme
A
+ v,B
e v,C
+ vDD
(1)
Here species A is generally a reactant in the gas phase and B, C,and D may be gaseous or nonvolatile liquid species. Reaction of A and B is assumed t o occur at the interior surface of the catalyst particles, which are suspended in the liquid medium. It can be shown that the overall mass transfer rate of species i from the gas phase or the bulk liquid t o the external surface of the catalyst pellet can be expressed as
where
ci.= {& ;
for gaseous species for nonvolatile liquid species
(3)
and 1
for gaseous species for nonvolatile liquid species (4)
Under steady state conditions, the dimensionless mass balance equation for component A, chosen as the key
0888-588519512634-3771$09.00/0 0 1995 American Chemical Society
3772 Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995 component, in a spherical catalyst pellet can be written as
A as v, = 0. A Taylor series expansion of f i )around y = yo gives
(5) together with the boundary conditions
Substitution of eqs 14 and 15 in eq 5 yields
-d _y - ~ at x = o dx
(16)
where x = rlL, y = CAICx,and f i )= R(C)IR(C*).The Thiele modulus is defined as
and the Damkohler number as subject to
From the relation between the diffusional flux for species A and that for species i inside the catalyst pellet, it is easily found
On the other hand, from the relation between transfer rate of species A and that for species i outside the catalyst pellet, we can obtain
GL(O) = 0
GL(1) = - G1(l) 3Da
(22a,b)
G;(O) = 0
Gi(1) = - G2(1) 3Da
(23a,b)
Integration of eq 16 with respect to x gives, after use of eq 20, Substituting eq 11 into eq 10 and eliminatingyi(l1, we can obtain
-
That is to say, yo is constant within the catalyst pellet as v, 0. Integration by parts of eq 17 gives, after use of eq 21a,
where yi = C,IC: ai = viDiCxIDeC$ and yi = v;M,CIIM,C? The overall effectiveness factor is defined as *
‘
I
_
.
Insertion of eq 25 in eq 21b yields
(13) It is extremely difficult t o obtain a general solution of the overall effectiveness factor even with numerical methods. Nevertheless, an approximate expression may be obtained with the perturbation and matching technique (Wedel and Luss, 1980) which is based on the knowledge of the asymptotic behavior of the solutions for small and large values of the Thiele moduli. When v, 0, the solution of eqs 5-7 can be determined by use of the regular perturbation series
-
where yo(x) is the dimensionless concentration of species
yo = 1 - D a f i , )
(26)
Integration of eq 25 gives
Insertion of eq 27 in eq 18 followed by integration by parts yields, after use of eq 22a,
By substituting eqs 27 and 28 into eq 22b, c2 is obtained
Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995 3773 tion of the key species a t the center of the catalyst pellet,
as following
y*, is equal to its equilibrium concentration for revers-
ible reaction or zero for irreversible reaction, i.e.,
Integration of eq 28 gives
G2(x)= -
y*
+ c3 &#Yo) f’(yo>X4 + $’(yo)cg2 1
(30)
Insertion of eqs 27 and 30 in eq 19 followed by integration by parts gives, after use of eq 23a,
i
yeq for reversible reaction as e 0 for irreversible reaction
-
00
for q -
03
(40)
The solutiony(e)can be represented by the perturbation series
and fcr) can be expanded by a Taylor series around Uo at a given point x
Substitution of eqs 30 and 31 into eq 23b gives, after several algebraic steps,
[fio);oiva) [f’(Yo)l2 +
+-f’b0)]c2
10 vlr0)12f”(Y0)
+
fi0)[f’(ro)l2
280
168 of
Inserting eqs 41 and 42 into eq 37 and equating the coefficients of the same power of Q, on both sides of the equation leads to
+
6Da +
f i o ) f’(Yo)}(32)
The corresponding boundary conditions are
U o = l at
120Da
e=O
(45)
at e = 0
(47)
By substituting eq 15 into eq 13, asymptotic solution for small Q, can be obtained as follows:
r = fro) + 0,q2+ a2q4+ 0(q6)
U,= 3Dav0
(33)
U,=O and v,=O as@--..
where
(34)
(48a,b)
Substitution of eq 41 into eq 13 gives
11 = plq-l
+ pzq-2 + ...
(49)
where
-
As Q, -, the singular perturbation analysis can be used to determine the behavior of 17. Introduction of the stretched coordinate = q(l -x )
(50)
Solving eqs 43 and 44 by use of the similar procedure used by Wedel and Luss (1980), we can obtain
(36)
into eqs 5-7 gives the following equations for the solutions close t o x = 1:
(51)
(37) 3Da (52)
--
In addition, as Q, the reaction rate is so large in comparison with the diffusion rate that the concentra-
The reason why the limits of integration in eqs 51 and 52 are y* to 1 is that the concentration of the key species a t the surface of the catalyst pellet, y(l), is equal to 1 as Q, =. Under this limiting condition, external transfer resistances are negligible in comparison with internal transfer resistances for a finite Damkohler number.
-
3774 Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995
Thus, the two leading terms in the asymptotic series expression of 7 (eq 49) for large p are determined by the following equations:
P1 = 3[2L; f i )dyI1" P2
18 1 -J Y* [Suo Y" 2 f i )
= - p1
a2 = ( p - UP1 + Pz a3 = -2a#
-
+
3P1pu2 4fi0)p3
+ 301p4 for q = 4 (63a-d)
(53) a4 = a#'
dyI1/' dUo - 9Da
+ 2p3p1 - 3fi0)p4 - 30J
(54)
a , = P1 The asymptotic solutions, eqs 33 and 49, contain most of the information extractable from the reactiondiffusion equation about the limiting values of 7 . For the common case in which a unique steady state solution exists, the two asymptotes are connected in a smooth fashion. In order to match these asymptotes and to obtain an approximation valid for all p, we use the following rational polynomial
+
a5 = -a+3 - 3p4P,
which is originated from the analytical expression of the effectiveness factor for first-order reaction occurring inside a slab. In the physical sense, any reaction with arbitrary kinetics and catalyst pellet geometry can be approximated to first-order reaction inside a slab to a certain extent, so it is possible for eq 55 to converge t o the proper asymptotic solutions for either small or large p and t o give a good estimation for 7 throughout the whole range of p. The coefficients al, az, ..., a4 are determined by expanding eq 55 in a Taylor series for both small and large p and requiring that the five leading terms in the first case and the three leading terms in the last case are equal to the first terms in eqs 33 and 49. This gives the following five relations (Yin, 1994): (56)
+
a3 = -3a# - 6&p2 10fi0)p3 + (18 O.6p)qp4 994' for q = 5 (64a-e)
+
+ 6fi0)p5 + (15
+ 0.6p)u$ + 90#'
by solving corresponding equations chosen from eqs 5660. Thus the approximate expression for 17 in which the number of terms can be equal to 3, 4, and 5 is finally obtained. One of the most important features of this expression is that the unknowns a,, az, ..., a4 are determined by a set of linear algebraic equations in a very simple and explicit form.
Results and Discussion In order to test the accuracy of the present approximate expression and to investigate the behavior of the deviation between the approximate and exact 7 , values of 7 obtained with eq 55 are first compared with those generated by the analytical expression of the overall effectiveness factor,
m=l
(57) 4
C m a m ( 5 m + 1 5 + 2p)p-"-'
= 900,
(58)
m=l
a , = P1 a2 = (p -
(59)
Up1 + Pz
(60)
where p = 3Da
+1
(61)
From eqs 56-60 five coefficients can be determined at most. Considering the proportionality of the information about the limiting values of 7 , we can obtain
a , = P1 a2 =
-2pP1
+ 3fi0)p2+ 3qp3
for q = 3
for first-order isothermal reaction with external transport resistances occurring inside a spherical catalyst pellet. The accuracy of the q-term expression is represented by the percent deviation d,, defined as
(62a-c)
where q4 and VE denote values computed by the q-term expression and eq 65, respectively. As an example, Figure 1 describes the dependence of 83 on p and Da. From Figure 1we can find that the maximum deviation as well as the corresponding p around which the maximum deviation occurs increases with the Damkohler number. This is due to that the value of p around which the asymptotic solution (eq 49) is tenable becomes larger and the gap between the asymptotic solutions valid for small and large p becomes wider as D a increases, as shown in Figure 2. From these figures, we can also find that the maximum deviation always occurs around intermediate values of p. Therefore, so long as the results in this range are of sufficient accuracy, the approximate expression can safely be used.
Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995 3775
-
Table 1. Maximum Deviations between Approximate and Numerical q for nth Order Reactions (Keq m, n~ = 0) nA
Da
(63)max
(b4)max
(65)max
0.5
0.1 1.0 5.0 0.1 1.0 5.0 0.1 1.0 5.0 0.1 1.0 5.0
-4.47 -1.25 1.12 1.23 -1.64 6.31 1.66 -1.32 6.03 1.94 -0.72 7.67
3.54 -1.54 0.23 2.69 -1.47 2.37 2.35 - 1.45 2.30 1.78 -1.67 1.86
-1.95 -1.25 -1.88 -0.68 -2.58 -0.79 -1.04 -2.70 -0.46 -1.76 -2.89 -0.90
1.5 2.0 -4',,4'1
1
'
'
""""
'
""""
"I
3.0
100
10
P Figure 1. Deviation between the estimated and exact r for the first-order reaction: 1,Da = 0.1; 2, Da = 1.0; 3, Da = 2.0; 4, Da = 5.0; 5,
Da = 7.0; 6, Da = 10.0
Table 2. Maximum Deviations between Approximate and Numerical q for Irreversible Reactions with Power-Law Kinetics (Kes =, UB = 0.5, y~ = 1.0,Da = 1.0)
-
nA
nB
(63)max
(64)max
(65)max
1.0 0.5 1.5 1.5 1.5 2.0
1.0 0.5 0.5 1.0 1.5 0.5
-2.51 -3.45 -1.86 -1.82 -1.63 -1.38
3.35 1.84 2.04 3.26 4.38 2.11
2.77 -2.89 -2.69 -2.73 2.86 -2.70
of species A can be determined by the following equation:
1
10
100
Q Figure 2. Asymptotic and exact r for first-order reaction: 1,Da = 0.1; 2, Da = 2.0; 3, Da = 10.0; -, exact values computed by eq 65; - - -, asymptotic values computed by eqs 33 and 49.
The validity of the general expression 55 is checked for reactions with kinetic expressions of the type generally encountered in catalysis, such as those of powerlaw and Langmuir-Hinshelwood. Values of the overall effectiveness factor obtained with eq 55 are compared with those generated by finite-difference method. The accuracy of the proposed expression is still represented by d,, defined by eq 66 in which V E is now computed by the finite-difference method. When the rate of reaction 1 is given by the powerlaw kinetic expression,
(67) the dimensionless rate of the reaction can be written as
(68) where
(69)
-
If Keq -, the reaction is irreversible. Otherwise, the reaction is reversible and the equilibrium concentration
In Table 1, the maximum deviations between the approximate and numerical 7 as a function of the Damkohler number for several nth-order reactions (namely K e q 00 and n~ = 0 in eq 67) are presented. It is shown that the approximate expression can yield results which are of sufficient accuracy for engineering purposes. Tables 2 and 3 present the maximum deviations between the estimated and numerical values of the overall effectiveness factor for irreversible and reversible reactions, respectively. It can be seen that our procedure is versatile in all cases investigated and the agreement between approximate and numerical results is surprisingly good. Many Langmuir-Hinshelwood type kinetic expressions can be reduced to the following form:
-
f i )= (1+ xys/(1+ xy>p
(71)
For this rate expression multiple steady state solutions exist for p > 1and sufficiently large X (Aris, 1975). The rational approximation 55 cannot represent this multiplicity and is expected to fail in such cases. To illustrate our procedure, the following cases are investigated: p = [0.5,1.0,1.5,2.01,X= [0.1,1.0,5.0,10.01, and Da = [0.1,1.0,5.0,10.01. In all cases the maximum deviations among the approximate and numerical 7 are below 4% and in most circumstances are around 2%, as shown in Table 4. It can be seen that our expression 55 is also adaptable for determination of the overall effectiveness factor for the reaction with LangmuirHinshelwood kinetics.
Conclusion
An explicit rational approximation of the overall effectiveness factor for gas-liquid-solid phase catalytic
3776 Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995 Table 3. Maximum Deviations between Approximate and Numerical 7 for Reversible Reactions with Power-Law Kinetics ( K = 5.0, Da = 5.0)
1 1
nB 0 0 0 1
1
1
nA
1 2
nc 1 1 2 1 1
nD 0 0 0 0
0.5
1
0.8
aB
ac -0.3 -0.8 -0.8 -0.8 -0.5
aD
1.0 1.0 1.0 5.0 1.0 0.1
0.1 5.0 1.0 5.0 10.0 5.0
-2.74 3.40 -1.07 -4.72 -1.22 4.01
2.90 1.41 -1.09 -2.98 -0.74 1.70
YC
YD
(63)rnax
(d4)rnax
(d5)rnz.x
-1.0
2.40 4.04 1.49 2.32 1.58
2.94 1.11 1.94 4.95 3.75
-3.25 -2.22 -2.98 -4.19 -3.77
-1.0 -0.8 -1.0
-0.8
Table 4. Maximum Deviations between Approximate and Numerical q for Irreversible Reactions with Langmuir-Hinshelwood Kinetics 0.5 0.5 1.0 1.0 1.5 2.0
YB
-1.14 -2.71 -1.70 -0.78 -1.47 -2.36
reaction with external resistances is presented. The method assumes that the asymptotic solutions of q valid for small and large p can be matched with a unique algebraic expression with several coefficients. These coefficients, however, are uniquely determined as a function of the parameters derived from q asymptotic expressions. The approximation method is versatile in its applicability to arbitrary rate expressions, and the results so obtained compare fairly well with analytical or numerical solutions whenever multiplicity is not present in the q-p graph. From all points of view, the fiveterm expression is the best and is to be preferred when very high accuracy is demanded. Nevertheless, the simple three-term and four-term expressions can also yield results which are of sufficient accuracy for design purposes. Although the analysis in this work was restricted to a spherical catalyst pellet, it can easily be extended t o other geometrical shapes of the catalyst.
Nomenclature a1 = effective gas-liquid interfacial area per unit reactor volume, cm2/cm3 up= external surface area of catalyst pellet per unit reactor volume, cm2/cm3 C, = local concentration of species i inside the catalyst pellet, mol/cm3 C,* = concentration of species i defined by eq 3, mol/cm3 C,, = concentration of species i in the gas phase, mol/cm3 C,, = concentration of species i in the liquid phase, mol/ cm3 C,, = concentration of species i at the external surface of catalyst pellet, mol/cm3 Da = Damkohler number 0: = effective diffusion coefficient of species i in the pores of the catalyst, cm2/s f = dimensionless reaction rate H , = solubility Coefficient of species i, cm3(liquid)/cm3(gas) k = reaction rate constant k , = mass transfer coefficient from gas to gas-liquid interface, c d s ki = mass transfer coefficient from gas-liquid interface to bulk liquid, c d s k , = mass transfer coefficient from bulk liquid to external surface of catalyst pellet, c d s Keq= chemical equilibrium constant L = radius of the catalyst pellet, cm M, = overall mass transfer coefficient of species i, defined by eq 4,s-l
1.0 1.0
-1.0 -1.2
n, = order of reaction with respect to species i N , = overall mass transfer rate of species i, defined by eq 2, mol/(cm3s) q = number of terms in approximate expression 55 r = radial coordinate, cm R = reaction rate, mol/(cm3s) x = dimensionless radial coordinate y* = dimensionless concentration of species A at the center of the catalyst pellet y = dimensionless concentration of species A yeq = dimensionless equilibrium concentration of species A y, = dimensionless concentration of species i Greek Symbols
6, = percent deviation between approximate and analytical or numerical 7,defined by eq 66 E~
= volume of catalyst per unit reactor volume, cm3/cm3
q = overall effectiveness factor Y, =
stoichiometric coefficient of species i
q = Thiele modulus
Literature Cited Aris, R. The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts; Clarendon Press: Oxford, 1975; Vol. 1, p 168. Bischoff, K. B. Effectiveness Factors for General Reaction Rate Forms. M C h E J . 1965, 11, 351. Chaudhari, R. V.; Ramachandran, P. A. Three Phase Slurry Reactors. M C h E J . 1980, 26, 177. Gonzo, E. E.; Gottifredi, J. C. Rational Approximations of Effectiveness Factor and General Diagnostic Criteria for Heat and Mass Transport Limitations. Catal. Rev.-Sci. Eng. 1983, 25, 119. Gottifredi, J. C.; Gonzo, E. E.; Quiroga, 0. D. Isothermal Effectiveness Factor-I Analytical Expression for Single Reaction with Arbitrary Kinetics. Slab Geometry. Chem. Eng. Sci. 1981a, 36, 705. Gottifredi, J. C.; Gonzo, E. E.; Quiroga, 0. D. Isothermal Effectiveness Factor-I1 Analytical Expression for Single Reaction with Arbitrary Kinetics, Geometry and Activity Distribution. Chem. Eng. Sci. 1981b, 36, 713. Ramachandran, P. A,; Chaudhari, R. V. Theoretical Analysis of Reaction of Two Gases in a Catalytic Slurry Reactor. Ind. Eng. Chem. Process Des. Deu. 1979, 18, 703. Wedel, S.; Luss, D. A Rational Approximation of the Effectiveness Factor. Chem. Eng. Commun. 1980, 7, 245. Yang, Jin; Li, Shaofen. Approximate Analytical Solution of Effectiveness Factor for porous Catalyst-I Approximate Expression. J . Chem. Znd. Eng. (China, Engl. Ed.) 1988, 3, 11. Yin, Qiuxiang. Kinetics of Gas-Liquid-Solid Phase Catalytic Synthesis of Methanol. Ph.D. Dissertation, Tianjin University, 1994. Received for review November 18, 1994 Accepted J u n e 1, 1995@ IE940687U
Abstract published in Advance ACS Abstracts, August 15, 1995. @
