IJRRAS 13 (3) ● December 2012 www.arpapress.com/Volumes/Vol13Issue3/IJRRAS_13_3_08.pdf 716 EFFECTIVENESS FACTOR FOR POROUS CATALYSTS WITH SPECIFIC EXOTHERMIC AND ENDOTHERMIC REACTIONS UNDER LANGMUIR-HINSHELWOOD KINETICS Gabriel Ateiza Adagiri 1 , Gutti Babagana 2 & Alfred Akpoveta Susu 3,* 1 Nordbound Integrated Engineering Services Ltd., P.O. Box 3111, Ikorodu, Lagos, Nigeria 2 Department of Chemical Engineering, University of Maiduguri, Borno State, Nigeria 3 Department of Chemical Engineering, University of Lagos, Lagos, Nigeria ABSTRACT The effectiveness factors of non-isothermal specific reactions of Langmuir-Hinshelwood expressions of real reacting systems were modeled through the specification of concentration and temperature profiles in the spherical catalyst pellet. The data obtained from Windes et al. [13] on the oxidation of formaldehyde over iron-oxide/molybdenum- oxide catalyst was used for the exothermic reaction, while vinyl acetate synthesis from the reaction of acetylene and acetic acid over palladium on alumina, as presented by Valstar et al. [14] was used for the endothermic reaction. The developed models were solved using orthogonal collocation numerical technique with third order semi-implicit Runge-Kutta method through FORTRAN programming. The results of the simulation of the experimental conditions for the exothermic reaction showed clearly that the effectiveness factor was at no point higher than unity, the same hold true for the endothermic reaction. However, as the temperature is reduced in the modeling effort, the exothermic effectiveness factors indicated an increasing maximum, as high as 98 for a Thiele modulus of about 0.06 where the reaction is diffusion free. This could be attributed to the opposing effects of the temperature and concentration profiles for the exothermic reaction where the concentration profile increased with increasing radius and the temperature profile showed the opposite effect. Keywords: Porous catalyst, Effectiveness factor, Nonisothermal reactions, Exothermic reaction, Endothermic reaction. Temperature profile, Concentration profile 1. INTRODUCTION The concept of effectiveness factor is an important one in heterogeneous catalysis and in solid fuel. The effectiveness factor is widely used to account for the interaction between pore diffusion and reactions on pore walls in porous catalytic pellets and solid fuel particles. The effectiveness factor is defined as the ratio of the reaction rate actually observed to the reaction rate calculated if the surface reactant concentration persisted throughout the interior of the particle, that is, no reactant concentration gradient within the particle. The reaction rate in a particle can therefore be conveniently expressed by its rate under surface conditions multiplied by the effectiveness factor. This concept was first developed mathematically by Thiele [1], and has since been extended by many other workers. Extensive investigation of analytical solutions and methods for the approximation of the effectiveness factor can be found in Aris [2,3]. The state of development of the theory up till the last decade has been summarized by Wijngaarden et al. [4]. Most of the chart and data available in open literature and other solutions are based on the simplified kinetics such as integer power-law kinetics, that is, first- or second-order reactions. Comparatively, attention given to the kinetics of complex expressions such as the Langmuir-Hinshelwood rate equation, has been very limited. Roberts and Satterfield [5] pointed out that over a narrow region of concentration, the Langmuir-Hinshelwood form may be well approximated by an integer-power equation. However, in a situation where resistance posed by diffusion inside the pellet is high, the reactant concentration term may decrease from the surface of the pellet down to a value approaching zero in the interior of the pellet. This concentration gradient will be large, and thus, necessitate the consideration of the effect of more complex rate forms for the effectiveness factor. The concentration gradient may be accompanied by temperature gradient due to the rate of chemical reaction for both exothermic and endothermic types. The temperature gradient for some practical cases may be negligible. In a situation in which the heat of reaction is large, Susu [6] pointed out that due to the presence of micropores and macropores, the effective thermal conductivities are low, and the resulting temperature gradient may be too large to be neglected. They may even be more significant than the concentration gradient in their effect on the reaction rate. Anderson [7] derived a criterion for negligible effect of temperature gradient, while Kubota et al. [8] derived a condition where both are not important. Even more worrisome are the theoretical predictions for exothermic reactions that indicated values of the effectiveness factors in excess of 100 for values of the Thiele modulus close to 0.1 [9], that is close to the region where diffusion is negligible. The question that immediately arises is: are such
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IJRRAS 13 (3) ● December 2012 www.arpapress.com/Volumes/Vol13Issue3/IJRRAS_13_3_08.pdf
716
EFFECTIVENESS FACTOR FOR POROUS CATALYSTS WITH
SPECIFIC EXOTHERMIC AND ENDOTHERMIC REACTIONS UNDER
reaction. Temperature profile, Concentration profile
1. INTRODUCTION
The concept of effectiveness factor is an important one in heterogeneous catalysis and in solid fuel. The
effectiveness factor is widely used to account for the interaction between pore diffusion and reactions on pore walls
in porous catalytic pellets and solid fuel particles. The effectiveness factor is defined as the ratio of the reaction rate
actually observed to the reaction rate calculated if the surface reactant concentration persisted throughout the interior
of the particle, that is, no reactant concentration gradient within the particle. The reaction rate in a particle can
therefore be conveniently expressed by its rate under surface conditions multiplied by the effectiveness factor. This
concept was first developed mathematically by Thiele [1], and has since been extended by many other workers.
Extensive investigation of analytical solutions and methods for the approximation of the effectiveness factor can be
found in Aris [2,3]. The state of development of the theory up till the last decade has been summarized by
Wijngaarden et al. [4].
Most of the chart and data available in open literature and other solutions are based on the simplified kinetics such
as integer power-law kinetics, that is, first- or second-order reactions. Comparatively, attention given to the kinetics
of complex expressions such as the Langmuir-Hinshelwood rate equation, has been very limited. Roberts and
Satterfield [5] pointed out that over a narrow region of concentration, the Langmuir-Hinshelwood form may be well
approximated by an integer-power equation. However, in a situation where resistance posed by diffusion inside the
pellet is high, the reactant concentration term may decrease from the surface of the pellet down to a value
approaching zero in the interior of the pellet. This concentration gradient will be large, and thus, necessitate the
consideration of the effect of more complex rate forms for the effectiveness factor.
The concentration gradient may be accompanied by temperature gradient due to the rate of chemical reaction for
both exothermic and endothermic types. The temperature gradient for some practical cases may be negligible. In a
situation in which the heat of reaction is large, Susu [6] pointed out that due to the presence of micropores and
macropores, the effective thermal conductivities are low, and the resulting temperature gradient may be too large to
be neglected. They may even be more significant than the concentration gradient in their effect on the reaction rate.
Anderson [7] derived a criterion for negligible effect of temperature gradient, while Kubota et al. [8] derived a
condition where both are not important. Even more worrisome are the theoretical predictions for exothermic
reactions that indicated values of the effectiveness factors in excess of 100 for values of the Thiele modulus close to
0.1 [9], that is close to the region where diffusion is negligible. The question that immediately arises is: are such
IJRRAS 13 (3) ● December 2012 Adagiri & al. ● Effectiveness Factor for Porous Catalysts
717
high values of the effectiveness factor really realizable, within feasible reaction parameters, even for exothermic
reactions? We can look for answers from the prediction of real reacting systems. Here, we start by looking at two of
such systems, one exothermic and the other endothermic.
In solving problems involving gradients of temperature and concentration in porous catalyst pellets, orthogonal
collocation method has been used by many authors since Villadsen and Stewart [10] and Villadsen [11] applied the
method to solve boundary value problems. Hlavacek et al. [12] discussed the application of the method in
comparison with linearization and difference method for various engineering problems including heat and mass
transfer in porous catalyst.
This research examines the effectiveness factor of real systems for both exothermic and endothermic reactions with
Langmuir-Hinshelwood rate equations using orthogonal collocation numerical method. These will however, be
limited to spherical pellets. The data obtained from Windes, et al.[13] in oxidation of formaldehyde over
commercial iron-oxide/molybdenum-oxide catalyst will be used in the exothermic study. For the endothermic study,
the data from vinyl acetate synthesis from the reaction of acetylene and acetic acid over palladium on alumina as
presented by Valstar, et al. [14] is chosen. The reactions are both carried out in fixed bed reactors, and are of
Langmuir-Hinshelwood type. Most of the theoretical models dealing with this topic have been devoted to theoretical
rate models. This work therefore focused on data of real reacting systems.
Besides, in the theory section, we will present a review of the effectiveness factor for various rate forms and
geometries to highlight the conflicting results of theoretical predictions in the literature. Furthermore, the theory of
orthogonal collocation will be presented in some detail in view of its application to the effectiveness factor in the
catalyst pellet for the solution of the mass and heat balance equations.
The resulting concentration and temperature profiles in the pellets will be presented and discussed. This will be used
to obtained effectiveness factors as a function of a modified Thiele modulus, Ø, for varying Arrhenius number, γ,
and the heat of reaction parameter, β, for the two reactions. The aim is to model non-isothermal effectiveness factor
of Langmuir-Hinshelwood rate equations of real reacting systems. The results will be compared with that of power
laws rates available in the literature.
2.THEORY
2.1Concept of Effectiveness Factor
Catalytic reactions take place on the exposed surface of a catalyst. Consequently, a higher surface area available for
the reaction yields a higher rate of reaction. It is therefore necessary to disperse an expensive catalyst on a support of
small volume and high surface area. However, use of such a supported catalyst in the form of a pellet is not without
its drawback. Reactants have to diffuse through the pores of the support for the reaction to take place, and therefore,
the actual rate can be limited by the rate at which the diffusing reactants reach the catalyst. This actual rate can be
determined in terms of intrinsic kinetics and pertinent physical parameters of the diffusion rate process. Thiele [1]
was one of the first to use the concept of an effectiveness factor. He defined the effectiveness factor as:
𝜂 =𝑔𝑙𝑜𝑏𝑎𝑙 𝑟𝑎𝑡𝑒
𝑖𝑛𝑡𝑟𝑖𝑛𝑠𝑖𝑐 𝑟𝑎𝑡𝑒 (2.1)
By definition, the global rate is simply the intrinsic rate multiplied by the effectiveness factor. In order to obtain an
expression for the effectiveness factor, conservation equations for the diffusion and reaction taking place in a pellet
are normally solved. The effectiveness factor has been popularly used for estimating the efficiency of catalytic
particles when a catalytic reactor is designed.
Wijngaarden et al. [4] pointed out that there are three main aspects in which the conversion rate inside the porous
catalyst depends. These are:
a) Micro properties of the catalyst pellet; the most important being pore size distribution, pore tortuosity,
diffusion rate of the reaction components in the gas phase, and diffusion rate of the reacting components
under Knudsen flow.
b) Macro properties which include size and shape of the pellet, and possible occurrence of anisotropy of the
catalyst pellet.
c) Reaction properties such as reaction kinetics, number of reactions involved, and complexity of the reaction
scheme under consideration.
The micro properties cannot be determined easily. Moreover, due to the complexity of diffusion of the reactions in a
solid matrix, the micro properties are usually accounted for by a lumped parameter, the so-called effective diffusion
co-efficient, De. For solid catalyst particles, this approach has proved to be very useful, provided that the particles
can be regarded as homogenous on a micro scale. Here it is assumed that it is possible to use the concept of an
effective diffusion co-efficient without too large error. Hence, the effect of micro properties is not usually of much
IJRRAS 13 (3) ● December 2012 Adagiri & al. ● Effectiveness Factor for Porous Catalysts
718
concern as it is assumed that the value De is known. The discussion is restricted usually to the impact of the macro
properties and reaction properties on the effectiveness factor.
2.2Calculation of Effectiveness Factor
Calculations of the effectiveness factor normally involve dimensionless numbers. Most common among these
numbers are: Thiele modulus (Ø), Arrhenius number, γ, and the heat of reaction parameter, β. Wijngaarden et al. [4]
has however introduced two other quantities called zeroth Aris number (An0) and first Aris number (An1). The
earlier ones are presented below.
2.2.1Thiele Modulus, Ø
When Thiele [1] developed the concept of effectiveness factor, he introduced a dimensionless number, called the
Thiele modulus to calculate the factor. This dimensionless modulus is defined, for first order reaction in a spherical
pellet, as:
Ø𝑇 = 𝑅 𝑅 𝐶𝐴 ,𝑠
𝐷𝑒𝐶𝐴 ,𝑠 (2.2)
where R is the distance from the centre of the catalyst pellet to the surface, 𝑅 𝐶𝐴,𝑠 is the conversion rate of
component A for surface conditions, 𝐷𝑒 is the effective diffusion of component A and 𝐶𝐴,𝑠 is the concentration of
component A at the outer surface of the catalyst pellet. These plots of the effectiveness factor versus Thiele modulus
Øt are available in the literature. As the Thiele modulus increases, the reaction becomes more limited by diffusion
and thus the effectiveness factor decreases. For high values of the Thiele modulus, the effectiveness factor is
inversely proportional to the Thiele modulus.
It can be seen that the Thiele modulus may be regarded as a measure for the ratio of the reaction rate to the rate of
diffusion. However, many definitions are used in the literature, in various attempts to generalize the term. Aris [15]
noticed that all the Thiele moduli for the first order reactions were of the following form for various shapes:
∅1 = 𝑋0 𝑘
𝐷𝑒 (2.3)
with k as the reaction rate constant and X0 a characteristic dimension. Aris [15] showed the curves of η versus Ø1
could be brought together in the low η region for all the catalyst shapes, if X0 is defined as:
𝑋0 =𝑉𝑃
𝐴𝑃 (2.4)
where VP and AP are the volume and external surface area, respectively, of the catalyst.
Plots of η versus Ø1 for several shapes are available in the literature. It can be seen that the curves coincide both in
the high and low η region. In the intermediate region the spread between the curves is largest. Wijngaarden et al. [4]
have observed that this spread is even larger for ring-shaped catalyst pellets
Generalization for the reaction kinetics has also been made. Petersen [16] has shown that for a sphere, a generalized
modulus can be postulated for nth-order kinetics.
∅1 = 𝑛+1
2𝑅
𝑘
𝐷𝑒𝐶𝐴,𝑠
𝑛−1 (2.5)
Using this generalized modulus, the effectiveness factor in the low η region (or for high Ø1) can be calculated from
𝜂 =3
Ø𝑠 (2.6)
Petersen [16] stated that a generalization of the Thiele modulus for the reaction order is also possible for other
shapes. For an infinite slab (or plate) he suggested, for the flow of region, the effectiveness factor could be
calculated by
𝜂 =1
Ø𝑃 (2.7)
with ØP being a generalized modulus, which follows from the following empirical correlation
Ø𝑃 =𝑛+2.5
3.5𝑅
𝑘
𝐷𝑒𝐶𝐴,𝑠
𝑛−1 (2.8)
This correlation should hold within 6%.
Rajadhyaksha and Vadusera [17] introduced a modified Thiele modulus for a sphere for nth
order kinetics, and
Langmuir-Hinshelwood kinetics with the rate equation.
𝑅 𝑐𝐴 =𝑘𝐶𝐴
1+𝐾𝐶𝐴 (2.9)
For nth
order kinetics
IJRRAS 13 (3) ● December 2012 Adagiri & al. ● Effectiveness Factor for Porous Catalysts
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Ø = 𝑛 𝑅 𝑘
𝐷𝑒𝐶𝐴,𝑠
𝑛−1 (2.10)
For Langmuir-Hinshelwood kinetics
Ø =1
1+𝐾𝐶𝐴 ,𝑠
𝐾𝐶𝐴 ,𝑠
ln 1+𝐾𝐶𝐴 ,𝑠 𝑅
𝑘
𝐷𝑒 (2.11)
It should be noticed that the modified modulus given in (2.5) and (2.10) are not in agreement.
A general expression for the modified Thiele modulus for an infinite slab was derived by Bischoff [18]:
∅𝑃 = 𝑅(𝐶𝐴,𝑠)𝑋 2 𝐷𝑒 (𝐶𝐴)𝑅(𝐶𝐴)𝐶𝐴 ,𝑠
0𝑑𝐶𝐴
1
2 (2.12)
If the effective diffusion coefficient 𝔇𝑒 is independent of the concentration CA, then for nth-order kinetics Equation
6.11 yields
Ø𝑃 = 𝑛+1
2𝑅
𝑘
𝐷𝑒𝐶𝐴,𝑠
𝑛−1 (2.13)
It should be noticed that again there is a discrepancy, this time between (2.8) and (2.13)
Other attempts have been made to arrive at modified Thiele modulus for different forms of reaction kinetics. For
example, Valdman and Hughes [19] have proposed a similar approximated expression for calculating the
effectiveness factor for Langmuir-Hinshelwood kinetics of type
𝑅 𝐶𝐴 =𝑘𝐶𝐴
1+𝐾𝐶𝐴 2 (2.14)
It should be noted, that in all of these cases, no actual reactions were indicated.
In addition to several empirical correlations, various numerical approximations have also been prosecuted [5]. Even
generalized numerical expression procedures are given, such as the collocation method of Finlayson [5], Ibanez [20]
and Namjoshi et al. [21].
2.2.2The Heat of Reaction Parameter, β
Another aspect of the problem under study here concerns catalyst particles with intra-particle temperature gradients.
In general, the temperature inside a catalyst pellet will not be uniform, due to heat effects of the reaction occurring
inside the catalyst pellet. The combination of the of two ordinary differential equations resulting from mass and heat
balances, with integration, will yield an expression that relate temperature inside the catalyst to the concentration:
𝑇
𝑇𝑠=
(−∆𝐻)𝐷𝑒𝐶𝐴 ,𝑠
𝜆𝑃𝑇𝑠 1 −
𝐶𝐴
𝐶𝐴 ,𝑠 (2.15)
where Ts is the surface temperature, (-∆H) the reaction enthalpy and λp the heat conductivity of the pellet.
For exothermic reactions, ΔH is negative, and the temperature inside the pellet is greater than the surface
temperature. The maximum temperature rise is obtained for complete conversion of the reactant, CA=0, that is: ∆𝑇𝑚𝑎𝑥
𝑇𝑠=
(−∆𝐻)𝐷𝑒𝐶𝐴 ,𝑠
𝜆𝑃𝑇𝑠 (2.16)
If the term β is defined as:
β =(−∆𝐻)𝐷𝑒𝐶𝐴 ,𝑠
𝜆𝑃𝑇𝑠 (2.17)
Then Equation 2.16 becomes: ∆𝑇𝑚𝑎𝑥
𝑇𝑠= β (2.18)
This parameter characterizes the potential for temperature gradient inside the particle.
2.2.3Arrhenius Number, γ
If the dependency of the conversion rate on the temperature is of the Arrhenius type, we can write [22]:
= 𝑒𝑥𝑝 +𝐸𝑎
𝑅𝑇𝑠 X
β 1−𝐶𝐴
𝐶𝐴 ,𝑠
1+β 1−𝐶𝐴
𝐶𝐴 ,𝑠 (2.19)
where ks is the reaction rate constant at the surface conditions, Ea is the energy of activation and R the ideal gas
constant. By defining
γ =𝐸𝑎
𝑅𝑇𝑠 (2.20)
𝑘
𝑘𝑠= 𝑒𝑥𝑝 +βγ X
β 1−𝐶𝐴
𝐶𝐴 ,𝑠
1+β 1−𝐶𝐴
𝐶𝐴 ,𝑠 (2.21)
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The extent to which the reaction rate depends on temperature can then be characterized as γ, defined in (2.20)
2.2.4Significance of the Dimensionless Quantities
Since the conversion rate depends on β and γ, the effectiveness factor will be defined by three parameters, namely,
β, γ and a Thiele modulus. For values of β larger than zero (exothermic reaction) an increase in the effectiveness
factor is found, since the temperature inside the catalyst pellet is higher than the surface temperature. For
endothermic reaction (β < 0), a decrease of the effectiveness factor is observed.
Criteria which determine whether or not intra-particle behavior may be regarded as isothermal, have been reviewed
by Mears [23], who gave as a criterion for isothermal operation: β𝛾 < 0.05𝑛 (2.22) where n is the reaction order. The temperature gradient inside the pellet must be taken into account if this criterion is
not fulfilled. For non-isothermal catalyst, many asymptotic solutions and approximations have been derived by
various authors [4, 24, 25].
2.3Orthogonal Collocation
The orthogonal collocation method has found widespread application in chemical engineering, particularly for
chemical reaction engineering. In the collocation method [26], the dependent variable is expanded in series.
𝑦 𝑥 = 𝑎𝑖𝑦𝑖(𝑥)𝑁+2𝑖=1 (2.23)
Suppose the differential equation is
𝑁 [𝑦] = 0 (2.24) Then the expansion is put into the differential equation to form the residual:
𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 = 𝑁 𝑎𝑖𝑦𝑖(𝑥)𝑁+2𝑖=1 (2.25)
In the collocation method, the residual is set to zero at a set of points called collocation points:
𝑁 𝑎𝑖𝑦𝑖 𝑥𝑗 𝑁+2𝑖=1 = 0, 𝑗 = 2, … . . , 𝑁 + 1 (2.26)
This provides N equations; two more equations come from the boundary conditions, giving N + 2 equations for N +
2 unknowns. This procedure is especially useful when the expansion is in a series of orthogonal polynomials, and
when the collocation points are the roots to an orthogonal polynomial, as first used by Lanczos [29,30]. A major
improvement was the proposal by Villadsen and Stewart [10] that the entire solution process be done in terms of the
solution at the collocation points rather than the coefficients in the expansion. Thus, Equation 2.24 would be