IJRRAS 13 (3) ● December 2012
www.arpapress.com/Volumes/Vol13Issue3/IJRRAS_13_3_08.pdf
EFFECTIVENESS FACTOR FOR POROUS CATALYSTS WITH
SPECIFIC EXOTHERMIC AND ENDOTHERMIC REACTIONS UNDER
LANGMUIR-HINSHELWOOD KINETICS
Gabriel Ateiza Adagiri1, Gutti Babagana2 & Alfred Akpoveta Susu3,*
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
1
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/molybdenumoxide 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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Adagiri & al. ● Effectiveness Factor for Porous Catalysts
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
(2.1)
𝑖𝑛𝑡𝑟𝑖𝑛𝑠𝑖𝑐 𝑟𝑎𝑡𝑒
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
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Adagiri & al. ● Effectiveness Factor for Porous Catalysts
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 (An 0) 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:
𝑅 𝐶𝐴 ,𝑠
Ø𝑇 = 𝑅
(2.2)
𝐷𝑒 𝐶𝐴 ,𝑠
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:
𝑘
∅1 = 𝑋0
(2.3)
𝐷𝑒
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 X 0 is defined as:
𝑉
𝑋0 = 𝑃
(2.4)
𝐴𝑃
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.
𝑛+1
∅1 =
2
𝑅
𝑘
𝐷𝑒
𝑛−1
𝐶𝐴,𝑠
(2.5)
Using this generalized modulus, the effectiveness factor in the low η region (or for high Ø 1) can be calculated from
3
𝜂=
(2.6)
Ø𝑠
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
1
𝜂=
(2.7)
Ø𝑃
with ØP being a generalized modulus, which follows from the following empirical correlation
Ø𝑃 =
𝑛+2.5
3.5
𝑅
𝑘
𝐷𝑒
𝑛−1
𝐶𝐴,𝑠
(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 =
th
𝑘 𝐶𝐴
(2.9)
1+𝐾𝐶𝐴
For n order kinetics
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IJRRAS 13 (3) ● December 2012
𝑘
Ø= 𝑛𝑅
𝐷𝑒
Adagiri & al. ● Effectiveness Factor for Porous Catalysts
𝑛−1
𝐶𝐴,𝑠
(2.10)
For Langmuir-Hinshelwood kinetics
Ø=
1
𝐾𝐶𝐴 ,𝑠
1+𝐾𝐶𝐴 ,𝑠
ln 1+𝐾𝐶𝐴 ,𝑠
𝑅
𝑘
(2.11)
𝐷𝑒
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]:
1
2
𝐶
∅𝑃 = 𝑅(𝐶𝐴,𝑠 )𝑋 2 0 𝐴 ,𝑠 𝐷𝑒 (𝐶𝐴 )𝑅(𝐶𝐴 ) 𝑑𝐶𝐴
(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
2
𝑘
𝑅
𝐷𝑒
𝑛−1
𝐶𝐴,𝑠
(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
𝑘 𝐶𝐴
𝑅 𝐶𝐴 =
(2.14)
2
1+𝐾𝐶𝐴
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)𝐷𝑒 𝐶𝐴 ,𝑠
1−
𝜆 𝑃 𝑇𝑠
𝐶𝐴
(2.15)
𝐶𝐴 ,𝑠
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, C A=0, that is:
(−∆𝐻)𝐷𝑒 𝐶𝐴 ,𝑠
∆𝑇𝑚𝑎𝑥
=
(2.16)
𝑇𝑠
𝜆 𝑃 𝑇𝑠
If the term β is defined as:
(−∆𝐻)𝐷𝑒 𝐶𝐴 ,𝑠
β=
(2.17)
Then Equation 2.16 becomes:
∆𝑇𝑚𝑎𝑥
=β
(2.18)
𝜆 𝑃 𝑇𝑠
𝑇𝑠
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
𝐶
β 1−𝐶 𝐴
𝐴 ,𝑠
(2.19)
𝐶
1+β 1− 𝐴
𝐶 𝐴 ,𝑠
where ks is the reaction rate constant at the surface conditions, E a is the energy of activation and R the ideal gas
constant. By defining
𝐸
γ= 𝑎
(2.20)
𝑅𝑇𝑠
𝑘
𝑘𝑠
= 𝑒𝑥𝑝 +βγ X
𝐶
β 1−𝐶 𝐴
𝐴 ,𝑠
(2.21)
𝐶
1+β 1− 𝐴
𝐶 𝐴 ,𝑠
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IJRRAS 13 (3) ● December 2012
Adagiri & al. ● Effectiveness Factor for Porous Catalysts
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 =
𝑁+2
𝑖=1 𝑎𝑖 𝑦𝑖 (𝑥)
(2.23)
Suppose the differential equation is
𝑁 [𝑦] = 0
Then the expansion is put into the differential equation to form the residual:
𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 = 𝑁
𝑁+2
𝑖=1 𝑎𝑖 𝑦𝑖 (𝑥)
(2.24)
(2.25)
In the collocation method, the residual is set to zero at a set of points called collocation points:
𝑁 𝑁+2
= 0, 𝑗 = 2, … . . , 𝑁 + 1
𝑖=1 𝑎𝑖 𝑦𝑖 𝑥𝑗
(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
evaluated at the collocation points:
𝑦 𝑥𝑗 = 𝑁+2
(2.27)
𝑖=1 𝑎𝑖 𝑦𝑖 𝑥𝑗 , 𝑗 = 1, … . . , 𝑁 + 2
and solved for the coefficients in terms of the solution at the collocation points:
−1
𝑎𝑖 = 𝑁+2
𝑦 𝑥𝑗 , 𝑖 = 1, . . . . , 𝑁 + 2
(2.28)
𝑖=1 𝑦𝑖 𝑥𝑗
Furthermore, if (2.23) is differentiated once and evaluated at all collocation points, the first derivative can be written
in terms of the values at the collocation points:
𝑑𝑦
𝑑𝑦 𝑖
−1
𝑥𝑗 = 𝑁+2
𝑦 𝑥𝑘
𝑥𝑗 , 𝑗 = 1, . . . . , 𝑁 + 2
(2.29)
𝑖,𝑘=1 𝑦𝑖 𝑥𝑘
𝑑𝑥
𝑑𝑥
or shortened to
𝑑𝑦
𝑥𝑗 = 𝑁+2
(2.30)
𝑖,𝑘=1 𝐴𝑗𝑘 𝑦 𝑥𝑘
𝑑𝑥
Rearranging, we have
−1 𝑑𝑦 𝑖
𝐴𝑗𝑘 = 𝑁+2
𝑥𝑗
(2.31)
𝑖=1 𝑦𝑖 𝑥𝑘
𝑑𝑥
Similar steps can be applied to the second derivative to obtain
𝑑2𝑦
𝑑𝑥 2
𝑥𝑗 =
𝐵𝑗𝑘 =
𝑁+2
𝑖=1
𝑁+2
𝑖,𝑘=1 𝐵𝑗𝑘
𝑦𝑖 𝑥𝑘
𝑦 𝑥𝑘 ,
2
−1 𝑑 𝑦 𝑖
𝑑𝑥2
(2.32)
𝑥𝑗
(2.33)
For the solution of the catalyst pellet problem, orthogonal collocation is applied at the interior points
𝑁+1
2
(2.34)
𝑖=1 𝐵𝑗 ,𝑖 𝐶𝑖 = ∅ 𝑅 𝐶𝑗 , 𝑗 = 1, … , 𝑁
and the boundary condition is solved for
𝐶𝑁+1 = 1
(2.35)
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IJRRAS 13 (3) ● December 2012
Adagiri & al. ● Effectiveness Factor for Porous Catalysts
The boundary condition at x=0 is satisfied automatically by trial function. After the solution has been obtained, the
effectiveness factor η is obtained by calculating
𝜂≡
1
𝑎 −1 𝑑𝑥
0 𝑅 𝑐 𝑥 𝑥
1
𝑎
−1 𝑑𝑥
0 𝑅 𝑐 1 𝑥
=
𝑁 +1
𝑖=1 𝑊 𝑗 𝑅 𝑐 𝑗 ,
𝑁 +1 𝑊 𝑅 1
𝑗
𝑖=1
(2.36)
3. MODEL DEVELOPMENT
To predict the influence of mass and heat transport in porous catalysts on the rate of heterogeneous reactions, it is
necessary to solve the differential mass balance of reaction mixture components together with the heat balance.
These balances will be based on a catalyst pellet of radius r shown in Figure 3.1 in a steady state non-isothermal
catalytic packed bed reactor. For the spherical pellet of voidage εp, diffusivity Dr , and effective thermal
conductivity 𝜆𝑠 , the mass and heat balance is presented below.
𝑟 + 𝛿𝑟
r
Figure 3.1: Material and energy balance for the solid phase in a single spherical particle
3.1Mass Balance
Mass 𝑖𝑛𝑝𝑢𝑡 𝑎𝑡 𝑟 − mass 𝑜𝑢𝑡𝑝𝑢𝑡 𝑎𝑡 𝑟 + 𝛿𝑟 + 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑑𝑢𝑒 𝑡𝑜 𝑐𝑕𝑒𝑚𝑖𝑐𝑎𝑙 𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛
rate of transfer from the pore of the fluid to the catalyst inner surface or rate of
+
absorption on catalyst inner surface
= 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛
(3.1)
𝑖𝑛𝑝𝑢𝑡 𝑎𝑡 𝑟 = −4𝜋εp 𝔇r
∂c s
∂r r
𝑜𝑢𝑡𝑝𝑢𝑡 𝑎𝑡 𝑟 + 𝛿𝑟 = −4𝜋εp 𝔇r
(3.2)
∂c s
(3.3)
∂r r+δr
∂q
𝑅𝑎𝑡𝑒 𝑜𝑓 𝑎𝑑𝑠𝑜𝑟𝑝𝑡𝑖𝑜𝑛 𝑜𝑛 𝑐𝑎𝑡𝑎𝑙𝑦𝑠𝑡 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 = 4𝜋r 2 ∂r
(3.4)
∂t
𝑅𝑎𝑡𝑒 𝑜𝑓 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑜𝑛 𝑠𝑜𝑙𝑖𝑑 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑜𝑣𝑒𝑟 𝑡𝑕𝑒 𝑡𝑖𝑚𝑒 𝑝𝑒𝑟𝑖𝑜𝑑 =
∂c
4𝜋r 2 εp ∂r s
(3.5)
∂t
2
𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑑𝑢𝑒 𝑡𝑜 𝑐𝑕𝑒𝑚𝑖𝑐𝑎𝑙 𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 = 4𝜋r ∂rηi R i (ci T)
(3.6)
Inserting Equation 3.2 to 3.6 into Equation 3.1 yields
∂cs
∂cs
∂q
∂cs
−4𝜋εp 𝔇r
− 4𝜋εp 𝔇r
− 4𝜋r 2 ∂r
+ (4𝜋r 2 εp ∂r
∂r r
∂r r+δr
∂t
∂t
∂c s
2
= 4𝜋r ∂rεp
(3.7)
∂T
Applying the mean value theorem of differential calculus to the first two terms on the left hand side of Equation
3.7 and taking limits as ∂r tends to zero, and then dividing by 4πr2∂r, we have:
εp
𝜕
𝜕𝑟
εp 𝔇r (
𝑟 2 𝔇r
∂ 2 𝑐 𝑠𝑖
∂𝑟 2
∂c si
− 𝜌𝑝
∂r
2 𝜕𝐶𝑠
+
𝑟 𝜕𝑟
𝜕𝑞
𝜕𝑡
) − 𝜌𝑝
− ηi R i (ci T) = εp
𝜕𝑞
𝜕𝑡
− ηi R i (ci T) =
∂c s
(3.8)
∂t
∂C
εp s
∂t
(3.9)
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IJRRAS 13 (3) ● December 2012
Adagiri & al. ● Effectiveness Factor for Porous Catalysts
Assuming steady state, we have:
∂2c
2 𝜕𝐶
𝜕𝑞
si
𝑠
εp 𝔇r
+
− 𝜌𝑝 = ηi R i (ci T)
∂𝑟 2
𝑟 𝜕𝑟
𝜕𝑡
Initial and boundary conditions are:
1) t=0; R>r>0; Cs=0
𝜕𝐶
2) r≥R
t>0
𝔇r 𝑠 = 𝐾𝑠 (𝐶𝑠𝑖 − 𝐶𝑓 )𝑟>𝑅
(3.11)
(3.12)
𝜕𝑟
∂C s
3)
(3.10)
= 0, t > 0
∂r r=0
(3.13)
Introducing dimensionless variables
i.
r2 = R2 𝛿
R
ii.
∂r = 1 𝜕𝛿
2𝛿 2
R2
2
∂r =
iii.
(3.14)
(3.15)
iv.
t=
v.
∂t =
vi.
cs =
vii.
τ=
viii.
∂τ =
ix.
Q∗i =
𝜕𝛿 2
4𝛿
τ
(3.16)
𝑍
Uf 𝑇
∂τ
Uf
c si
(3.17)
𝑍𝑇
(3.18)
(3.19)
c0
tU f
(3.20)
ZT
∂tU f
(3.21)
ZT
q ∗i
(3.22)
q ∗0i
Introducing the dimensionless variables into Equation 3.10, we have:
𝑐0 ∂ 2 c s
ℰ𝑝 𝔇𝑟
ℰ𝑝 𝔇𝑟
ℰ𝑝 𝔇𝑟
+
∂𝛿 2
𝑅2
4𝛿
4𝛿𝑐0 ∂ 2 c s
𝑅 2 ∂𝛿 2
4𝑐0 𝛿 ∂ 2 c s
𝑅2
∂𝛿 2
2𝑐0 ∂c s
−
1
𝑅 ∂𝛿
𝑅 ∂2
1
2𝛿 2
+
+
4𝑐0 ∂c s
𝑅 2 ∂𝛿
1 𝜕𝑐𝑠
𝛿 𝜕𝛿
Multiply both sides by
∂𝛿 2
𝑈𝑓
+
1 ∂c s
𝛿 ∂𝛿
𝑐0 𝜕𝑐𝑠
𝑍𝑇
𝜕𝑧
𝑞 0 𝜕𝑄 ∗ 𝑈 𝑓 𝑐0 𝜕c s
𝑐0 𝜕c s
𝑍𝑇 𝜕𝑧
𝑞 0 𝜕𝑄 ∗ 𝑈 𝑓 𝑐0 𝜕c s
−
−
𝑐0 𝜕c s
𝑍𝑇
𝜕𝑧
= 𝜂𝑅(𝑐, 𝑇)
(3.23)
= 𝜂𝑅(𝑐, 𝑇)
(3.24)
= 𝜂𝑅(𝑐, 𝑇)
(3.25)
𝑅 2 𝑍𝑇
𝔇𝑟 𝑐0 𝑈 𝑓
We have:
ℰ𝑝 4𝑍𝑇 𝛿 ∂ 2 c s
𝑞 0 𝜕𝑄 ∗ 𝑈 𝑓 𝑐0 𝜕𝑐𝑠
− 𝜌𝑝
𝑅 2 𝑞 0∗ 𝜕𝑄 ∗ 𝜕c s
𝔇𝑟 𝑐0 𝜕c s 𝜕τ
=
𝑅 2 𝑍𝑇
𝔇𝑟 𝑐0 𝑈 𝑓
𝜂𝑅(𝑐, 𝑇)
(3.26)
Thus, we have:
𝛼1 =
ℰ𝑝 4𝑍𝑇 𝛿
(3.27)
𝑈𝑓
𝑅 2 𝑞 0∗ 𝜕𝑄 ∗
𝛼2 = 𝜌𝑝
(3.28)
𝔇𝑟 𝑐0 𝜕c s
𝑅 2 𝑍𝑇
𝛼3 = 𝔇
𝛼1
∂2 cs
∂𝛿 2
(3.29)
𝑟 𝑐0 𝑈 𝑓
+
1 ∂c s
𝛿 ∂𝛿
− 𝛼2
𝜕c s
𝜕τ
= 𝛼3 𝜂𝑅(𝑐, 𝑇)
(3.30)
Assume mass transfer resistance is negligible, we have:
∂2c
1 ∂c
s
s
𝛼1
+
= 𝛼3 𝜂𝑅(𝑐, 𝑇)
∂𝛿 2
𝛿 ∂𝛿
Initial and boundary conditions are:
(1)
(3.31)
cs = 0; τ ≤ 0; 1 ≥ 𝛿 ≥ 0
(3.32)
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IJRRAS 13 (3) ● December 2012
4
(2)
∂c s
Sh ∂𝛿
∂c s
(3)
∂𝛿
𝛿=1
𝛿=0
Adagiri & al. ● Effectiveness Factor for Porous Catalysts
= cf − cs
(3.33)
𝛿=1
= 0; τ > 0
(3.34)
3.2Heat Balance
𝐻𝑒𝑎𝑡 𝑖𝑛𝑝𝑢𝑡 𝑎𝑡 𝑟 − 𝑕𝑒𝑡𝑎 𝑜𝑢𝑡𝑝𝑢𝑡 𝑟 + 𝛿𝑟 + 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑑𝑢𝑒 𝑡𝑜 𝑐𝑕𝑒𝑚𝑖𝑐𝑎𝑙 𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛
+ 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑎𝑑𝑠𝑜𝑟𝑝𝑡𝑖𝑜𝑛𝑜𝑛 𝑐𝑎𝑡𝑎𝑙𝑦𝑠𝑡 𝑖𝑛𝑛𝑒𝑟 𝑠𝑢𝑟𝑓𝑎𝑐𝑒
= 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛
𝜕𝑦
𝑖𝑛𝑝𝑢𝑡 𝑎𝑡 𝑟 = −4𝜋𝑟 2 𝜀𝑝 𝜆𝑠
`
𝜕𝑥 𝑟
𝑜𝑢𝑡𝑝𝑢𝑡 𝑎𝑡 𝑟 + 𝛿𝑟 = −4𝜋𝑟 2 𝜀𝑝 𝜆𝑠
𝜕𝑦
(3.35)
(3.36)
(3.37)
𝜕𝑥 𝑟+𝛿𝑟
𝑟𝑎𝑡𝑒 𝑜𝑓 𝑎𝑑𝑠𝑜𝑟𝑝𝑡𝑖𝑜𝑛 𝑜𝑛 𝑐𝑎𝑡𝑎𝑙𝑦𝑠𝑡 𝑖𝑛𝑛𝑒𝑟 𝑠𝑢𝑟𝑓𝑎𝑐𝑒
𝜕𝑞
= 4𝜋𝑟 2 𝜕𝑟 Δ𝐻𝜌𝑝
(3.38)
𝜕𝑡
𝑟𝑎𝑡𝑒 𝑜𝑓 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑜𝑛 𝑠𝑜𝑙𝑖𝑑 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑜𝑣𝑒𝑟 𝑡𝑕𝑒 𝑡𝑖𝑚𝑒 𝑝𝑒𝑟𝑖𝑜𝑑
𝜕𝑇
= 4𝜋𝑟 2 𝜕𝑟𝜀𝑝 𝜌𝑠 𝐶𝑝,𝑠 𝑠
(3.39)
𝜕𝑡
𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑑𝑢𝑒 𝑡𝑜 𝑐𝑕𝑒𝑚𝑖𝑐𝑎𝑙 𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 = 4𝜋𝑟 2 𝜕𝑟 Δ𝐻r ηi R i 𝑐𝑖 , 𝑇
(3.40)
Assuming steady state and negligible heat transfer resistances, we have
𝜕𝑦
𝜕𝑦
−4𝜋𝑟 2 𝜀𝑝 𝜆𝑠
— 4𝜋𝑟 2 𝜀𝑝 𝜆𝑠
𝜕𝑥 𝑟
𝜕𝑥 𝑟+𝛿𝑟
2
= 4𝜋𝑟 𝜕𝑟 Δ𝐻r ηi R i 𝑐𝑖 , 𝑇
(3.41)
Applying the mean value theorem of differential calculus to the first two terms on the left hand side of Equation
3.41, and taking limit as 𝛿𝑟 approaches 0, and dividing by 4𝜋𝑟 2 𝜕𝑟, we have:
𝜕
𝜕𝑇
𝜀𝑝
𝑥 2 𝜆𝑠 𝑠 = −Δ𝐻r ηi R i 𝑐𝑖 , 𝑇
(3.42)
𝜕𝑡
𝜕𝑟
Initial and boundary conditions are:
i.
𝑡 ≤ 0, 𝑅 > 𝑟 > 0, 𝑇𝑠 = 𝑇𝑤 = 𝑇0
(3.43)
𝜕 𝑇𝑠
ii.
𝑟 = 𝑅, 𝑡 > 0; −𝜆𝑠
= 𝛼𝑠 𝑇𝑠 − 𝑇𝑓
(3.44)
𝑟 = 0, 𝑡 > 0;
iii.
𝜕𝑇𝑠
𝜕𝑟
𝜕𝑟 𝑟=0
=0
(3.45)
Introducing the following dimensionless variables:
i.
𝑟 2 = 𝑅2 𝛿
𝑅
ii.
𝜕𝑟 = 1/2 𝜕𝛿
2𝛿
𝑅
𝜕𝑟 2 =
iii.
𝜏
iv.
𝑡=
v.
𝜕𝑡 =
vi.
𝑇𝑠 =
vii.
𝜏=
viii.
𝜕𝜏 =
ix.
𝑄1∗ =
4𝛿
(3.46)
(3.47)
𝜕𝛿 2
(3.48)
𝑍
𝑈𝑓 𝑇
𝜕𝜏
𝑈𝑓
𝑇𝑠
(3.49)
𝑍𝑇
(3.50)
(3.51)
𝑇0
𝑡𝑈 𝑓
(3.52)
𝑍𝑇
𝜕𝑡 𝑈 𝑓
(3.53)
𝑍𝑇
𝑞∗
(3.54)
𝑞0
Introducing these dimensionless variables into Equation 3.42, we have:
4𝜀 𝑝 𝜆 𝑠
𝜕 2 𝑇𝑠
𝑅2
𝜕𝛿 2
+
1 𝜕𝑇𝑠
(3.55)
Subject to initial and boundary conditions:
i.
𝑇𝑠 = 1; 𝜏 ≤ 0; 1 ≥
(3.56)
4
ii.
iii.
Let 𝛼4 =
=
1
−Δ𝐻𝑟 𝜂𝑖 𝑅𝑖 𝑐𝑖 , 𝑇𝑖
𝛿 𝜕𝛿
𝑇0
𝜕𝑇𝑠
𝑁𝑢 𝜕𝛿 𝛿=1
𝜕𝑇𝑠
𝜕𝛿 𝛿=0
4𝜀 𝑝 𝜆 𝑠
= 𝑇𝑓 − 𝑇𝑠
(3.57)
𝛿=1
= 0; 𝜏 > 0
(3.58)
𝑅2
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IJRRAS 13 (3) ● December 2012
𝛼4
𝜕 2 𝑇𝑠
𝜕𝛿 2
1 𝜕𝑇
+
1
=
𝛿 𝜕𝛿
Adagiri & al. ● Effectiveness Factor for Porous Catalysts
−Δ𝐻𝑟 𝜂𝑖 𝑅𝑖 𝑐𝑖 , 𝑇𝑖
𝑇0
(3.59)
4. ORTHOGONAL COLLOCATION TECHNIQUE
Orthogonal collocation technique is applied to the resulting mass and heat balance equations of Equations 3.31 and
3.59, respectively, as follows.
4.1Application of Orthogonal Numerical Technique on Mass Balance Equation
To apply orthogonal numerical technique, the first and second spatial derivatives at any interior collection point can
be expressed in matrix notation as:
∂c s
= 𝑁+1
(4.1)
𝐾=1 𝐴𝑗 ,𝑘 cs 𝑖,𝑘
∂Z
∂2cs
= 𝑁+1
𝐾=1 𝐵𝑗 ,𝑘 cs 𝑖,𝑘
Substituting Equations 4.1 and 4.2 into Equation 3.31, we have:
(4.2)
∂Z 2
𝛼1
𝑁+1
𝐾=1 𝐵𝑗 ,𝑘 cs 𝑖,𝑘
+
1
𝑁+1
𝐾=1 𝐴𝑗 ,𝑘 cs 𝑖,𝑘
𝛿𝑗
= 𝛼3 𝜂𝑅(𝑐, 𝑇)
(4.3)
Expanding Equation 4.3:
𝛼1
𝑁+1
𝐾=1 𝐵𝑗 ,𝑘 cs 𝑖,𝑘
+ 𝐵𝑗 ,𝑘 cs 𝑖,𝑁+1 +
= 𝛼3 𝜂𝑅(𝑐, 𝑇)
Factorizing like terms, we have:
𝛼1
𝑁
𝐾=1 𝐵𝑗 ,𝑘
+
1
𝑁+1
𝐾=1 𝐴𝑗 ,𝑘
𝛿𝑗
1
𝑁+1
𝐾=1 𝐴𝑗 ,𝑘 cs 𝑖,𝑘
𝛿𝑗
+ 𝐴𝑗 ,𝑘 cs 𝑖,𝑁+1
(4.4)
cs 𝑖,𝑘 + 𝐵𝑗 ,𝑁+1 +
1
𝛿𝑗
𝐴𝑗 ,𝑁+1 cs 𝑖,𝑁+1
= 𝛼3 𝜂𝑅(𝑐, 𝑇)
To substitute for cs 𝑖,𝑁+1 , the concentration at the surface of the pellets, we use Equation 3.33:
4
Sh
𝑁+1
𝐾=1 𝐴𝑁+1,𝑘
+ cs 𝑖,𝑘 = cf 𝑖,𝑗 − cs 𝑖,𝑁+1
Therefore,
cs 𝑖,𝑁+1 = Φ1 cf 𝑖,𝑗 −
where
1
Φ1 = 4
1+
Sh
𝛼1
(4.5)
𝑁+1
𝐾=1 𝐴𝑁+1,𝑘
(4.6)
+ cs 𝑖,𝑘
(4.7)
(4.8)
𝐴𝑁 +1, 𝑁 +1
𝑁
𝐾=1 𝐵𝑗 ,𝑘
+
1
𝑁
𝐾=1 𝐴𝑗 ,𝑘
𝛿𝑗
cs 𝑖,𝑘 + 𝐵𝑗 ,𝑁+1 +
1
𝛿𝑗
𝐴𝑗 ,𝑁+1
𝑁+1
𝐾=1 𝐴𝑁+1,𝑘
Φ1 cf 𝑖,𝑗 −
+ cs 𝑖,𝑘
= 𝛼3 𝜂𝑅(𝑐, 𝑇)
(4.9)
Thus we have,
𝑁
𝛼1
𝐵𝑗 ,𝑘
𝐾=1
1
+
𝛿𝑗
= 𝛼3 𝜂𝑅(𝑐, 𝑇)
Therefore,
𝐹 𝑗 = 𝛼1
𝑁
𝐴𝑗 ,𝑘 − 𝐵𝑗 ,𝑁+1 𝐴′𝑁+1,𝑘 −
𝐾=1
1
1
𝐴
𝐴′
c
+ 𝐵𝑗 ,𝑁+1 Φ1 + 𝐴𝑗 ,𝑁+1 Φ1 cs 𝑖,𝑘
𝛿𝑗 𝑗 ,𝑁+1 𝑁+1,𝑘 s 𝑖,𝑘
𝛿𝑗
(4.10)
𝑁
𝐾=1 𝐵𝑗 ,𝑘
𝛼3 𝜂𝑅(𝑐, 𝑇)
+
1
𝛿𝑗
𝑁
𝐾=1 𝐴𝑗 ,𝑘
− 𝐵𝑗 ,𝑁+1 𝐴′𝑁+1,𝑘 −
1
𝛿𝑗
𝐴𝑗 ,𝑁+1 𝐴′𝑁+1,𝑘 cs 𝑖,𝑘 + 𝐵𝑗 ,𝑁+1 Φ1 +
1
𝛿𝑗
𝐴𝑗 ,𝑁+1 Φ1 cf 𝑖,𝑗 −
(4.11)
4.2Application of Orthogonal Numerical Technique on Heat Balance Equation
Using orthogonal numerical technique as in mass balance equation, we have:
𝛼4
𝑁+1
𝑘=1 𝐵𝑗 ,𝑘 𝑇𝑠 𝑖,𝑘
+
1
𝛿𝑗
𝑁+1
𝑘=1 𝐴𝑗 ,𝑘 𝑇𝑠 𝑖,𝑘
=
1
𝑇0
−Δ𝐻𝑟 𝜂𝑖 𝑅𝑖 𝑐𝑖 , 𝑇𝑖
(4.12)
Expanding Equation 4.12, we have:
𝛼4
𝑁+1
𝑘=1 𝐵𝑗 ,𝑘 𝑇𝑠 𝑖,𝑘
+ 𝐵𝑗 𝑁+1 𝑇𝑠 𝑁+1 +
1
𝛿𝑗
𝑁
𝑘=1 𝐴𝑗 ,𝑘 𝑇𝑠 𝑖,𝑘
+ 𝐴𝑗 𝑁+1 𝑇𝑖,𝑁+1
=
1
𝑇0
−Δ𝐻𝑟 𝜂𝑖 𝑅𝑖 𝑐𝑖 , 𝑇𝑖
(4.13)
Factorizing like terms we have:
𝛼4
𝑁
𝑘=1 𝐵𝑗 ,𝑘
+
1
𝛿𝑗
𝑁
𝑘=1 𝐴𝑗 ,𝑘
𝑇𝑠 𝑖,𝑘 + 𝐵𝑗 𝑁+1 +
1
𝛿𝑗
𝐴𝑗 ,𝑁+1 𝑇𝑠 𝑖,𝑁+1 =
Substituting for 𝑇𝑠 𝑖,𝑁+1 , using Equation 3.57, we have:
724
1
𝑇0
−Δ𝐻𝑟 𝜂𝑖 𝑅𝑖 𝑐𝑖 , 𝑇𝑖
(4.14)
IJRRAS 13 (3) ● December 2012
4
𝑁𝑢
𝑁+1
𝑘=1 𝐴𝑁+1 𝑇𝑠 𝑖,𝑘
= 𝑇𝑓
Thus, we have:
𝑇𝑠 𝑖,𝑁+1 = 𝛷2 𝑇𝑖,𝑗 −
Therefore,
𝑇𝑠 𝑖,𝑁+1 = 𝛷2 𝑇𝑓 −
𝛼1
− 𝑇𝑠 𝑖,𝑁+1
(4.15)
N+1 ′′
k=1 A N+1,k Ts i,k
(4.16)
𝑁+1 ′′
𝑘=1 𝐴𝑁+1,𝑘 𝑇𝑠 𝑖,𝑘
𝑖,𝑗
where
𝛷2 =
𝑖,𝑗
Adagiri & al. ● Effectiveness Factor for Porous Catalysts
(4.17)
1
(4.18)
4
𝐴
𝑁𝑢 𝑁 +1,𝑁 +1
1+
𝑁
𝑘=1 𝐵𝑗 ,𝑘
+
1
𝛿𝑗
𝑁
𝑘=1 𝐴𝑗 ,𝑘
𝑇𝑠 𝑖,𝑘 + 𝐵𝑗 𝑁+1 +
= −𝛥𝐻𝑟 𝜂𝑖 𝑅𝑖 𝑐𝑖 , 𝑇𝑖
Thus we have,
𝛼4
𝑁
𝑘=1 𝐵𝑗 ,𝑘
+
1
𝛿𝑗
1
𝛿𝑗
𝐴𝑗 ,𝑁+1
𝛷2 𝑇𝑓
𝑖,𝑗
−
1
′′
𝑘=1 𝐴𝑁+1,𝑘 𝑇𝑠 𝑖,𝑘
(4.19)
𝑁
𝑘=1 𝐴𝑗 ,𝑘
− 𝐵𝑗 𝑁+1 𝐴′′𝑁+1,𝑘 −
+ 𝐵𝑗 ,𝑁+1 𝛷2 +
1
𝛿𝑗
1
𝛿𝑗
𝐴𝑗 ,𝑁+1 𝐴′′𝑁+1,𝑘 𝑇𝑠 𝑖,𝑘
𝐴𝑗 ,𝑁+1 𝛷2 𝑇𝑓
= −𝛥𝐻𝑟 𝜂𝑖 𝑅𝑖 𝑐𝑖 , 𝑇𝑖
𝑖,𝑗
(4.20)
Therefore we have,
𝐹𝑗 = 𝛼1
𝑁
𝑘=1 𝐵𝑗 ,𝑘
+
1
𝛿𝑗
𝑁
𝑘=1 𝐴𝑗 ,𝑘
− 𝐵𝑗 𝑁+1 𝐴′′𝑁+1,𝑘 −
1
𝛿𝑗
𝐴𝑗 ,𝑁+1 𝐴′′𝑁+1,𝑘 𝑇𝑓
𝑖,𝑘
— 𝛥𝐻𝑟 𝜂𝑖 𝑅𝑖 𝑐𝑖 , 𝑇𝑖
(4.22)
4.3Computer Simulation Flow Charts
FORTRAN programs were used to solve the balance equations in order to obtain the concentration and radial
profiles in the pellet. Based on the specified concentration and temperature profiles, another program was used to
obtain the effectiveness factor as a function of Thiele modulus of Equation 2.11. The algorithms used are given in
Figures 4.1 and 4.2, respectively.
4.4Subroutine Programs
The applied subroutines in the main program are JCOBI, DFOPR, STIFF, SIRK, BACK, LU, FUN, DFUN and
OUT. The subroutine FUN, DFUN and OUT are external subroutines while SIRK3, BACK, LU are internal STIFF3
𝛼 ,𝛽
subroutines. JCOBI SUBROUTINE calculates the zeros 𝑃𝑤 𝑥 and also the three first derivatives of the node
polynomial. SUBROUTINE DFOPR subroutine evaluates discretization matrices and Gaussian Quadratic weight
normalized to sum1. SUBROUTINE BACK finds the solution of Linear Equation by back substitution after
decomposition.
SUBROUTINE LU performs triangular decomposition by Gaussian elimination with partial pivoting. The program
is for decomposing a matrix A to a lower and upper triangular form A=LU. SUBROUTINE SIRK3 performs singlestep semi-implicit integration. And SUBROUTINE STIFF3 is used to solve the ODEs resulting from the conversion
of partial differential equation to ODEs. It solves the semi implicit Runge-Kuta method.
725
IJRRAS 13 (3) ● December 2012
Adagiri & al. ● Effectiveness Factor for Porous Catalysts
Start
Read specifications, radial diffusion, mass
transfer, thermal conductivity, initial
concentration, initial temperature
Read the exponent of JACOBI
polynomials
Compute mass pellet, heat pellet nos,
α1, α2, α3, etc
Initialise concentrations, temperature and
all variable in common statement
Use JCOBI subroutine to determine the
roots in radial direction
Calculate error multipliers
X
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IJRRAS 13 (3) ● December 2012
Adagiri & al. ● Effectiveness Factor for Porous Catalysts
X
Set up and solve the ODES in time, & distance in STIFF3.
ODE are set up in subroutine FUN. The JACOBI matrix is
evaluated in subroutine DFUN. Subroutine OUT prints the
computational results. FUN, DFUN & OUT are external
subroutines, SIRK3, BACK, LU, ARE INTERNAL STIFF3
SUBTOUTINE
Print dimensionless radius,
dimensionless concentration and
temperature
Stop
Figure 4.1: Dimensionless concentration and temperature flow chart
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IJRRAS 13 (3) ● December 2012
Adagiri & al. ● Effectiveness Factor for Porous Catalysts
Start
Read Pellet
specification
Read temperature,
concentration, radius
Compute Thiele modulus
Compute effectiveness factor as a
function of Thiele modulus
Print Thiele Modulus,
Effectiveness factor.
Stop
Figure 4.2: Thiele modulus and Effectiveness factor flow chart
5. DATA PROCUREMENT
This section presents the summary of chemical reaction data used in this work. It will also indicate modifications
that were made to the original work in order to suit the purpose of this work.
5.1Exothermic Reaction
The exothermic reaction chosen was from the pilot plant experiment of Windes et al. [13]. It involves the partial
oxidation of formaldehyde to carbon monoxide and water. This is a consecutive reaction in the partial oxidation of
methanol to formaldehyde over iron-oxide/molybdenum oxide catalyst.
The reaction was favored due to its high exothermic nature and the simplicity of the Langmuir-Hinshelwood rate
suits the present investigation. The chemical reaction and the data are as follows.
𝐶𝐻2 𝑂 +
−𝑟1 =
1
2
𝑂2
𝑘1
𝐶𝑂 + 𝐻2 𝑂
(5.1)
0.5
𝑘 1 𝐶𝐶𝐻
2𝑂
(5.2)
0.5
1+0.2𝐶𝐶𝐻
2𝑂
728
IJRRAS 13 (3) ● December 2012
𝑘1 = 5.4 X 105 exp⁡(−
66944
𝑅𝑔 𝑇
Adagiri & al. ● Effectiveness Factor for Porous Catalysts
)
(5.3)
Table 4.1: Reactor geometry, kinetic and transport parameters and operating conditions used in the exothermic
simulation
Parameter
Dimension
Value
L
[m]
0.7
dt
[m]
0.0266
dpv
[m]
0.0046
Ε
[ ]
0.5
us
[m/s]
2.47
ρf
[kg/m3]
1.018
cpf
[J/(kg.K)]
952
Tin
[K]
517
Tw
[K]
517
-ΔH
[J/mol]
158700
Peh
[ ]
8.6
Pem
[ ]
6.6
Bi
[ ]
5.5
Uw
[W/(m2.K)]
220
kf
[m/s]
0.25
hfs
[W/m2.K]
400
De
[m2/s]
4.9 X 10-6
λp
[W/m.K]
2
[mole/m3]
34
𝐶𝑂02
3
0
[mole/m
]
1.74
𝐶𝐶𝐻2 𝑂
5.2Endothermic Reaction
The work of Valstar et al. [14] was adopted for the endothermic study. It is the synthesis of vinyl formaldehyde
from acetylene and acetic acid over palladium catalyst. The chemical reaction taking place and the data provided
and adopted in this work are as follows:
𝐶2 𝐻2 + 𝐶𝐻3 𝐶𝑂𝑂𝐻
𝑅=
𝑅
𝐶𝐻3 𝐶𝑂2 𝐶𝐻𝐶𝐻2
(5.4)
𝑘 ∞ exp ⁡
(−𝐸/𝑅𝑔 𝑇)𝑃 𝐶 2 𝐻 2
∆𝐻 1
∆𝑆 1
exp
𝑃 𝐶𝐻 3 𝐶𝑂𝑂𝐻 +𝐾1 𝑝 𝐶𝐻 3 𝐶𝑂 2 𝐶𝐻𝐶 𝐻 2
𝑅𝑔𝑇
𝑅𝑔
(5.5)
1+exp −
The data on the reaction rate expression, the reactor geometry, transport parameters and operating conditions are
listed in Table 4.2.
Table 4.2: Reactor geometry, kinetic and transport parameters and operating conditions used in the endothermic
simulation
Parameter
Dimension
Value
L
dt
dp
Ε
us
ρf
cpf
Tin
Tw
ΔH
Pehr
Pemr
Bi
Ea
ΔS1
k∞
K1
[m]
[m]
[m]
[ ]
[m/s]
[kg/m3]
[J/(kg.K)]
[K]
[K]
[J/mol]
[ ]
[ ]
[ ]
[kJ/(mole)]
[J/mole.K]
[mole/m3 cat s atm-1]
[atm-1]
1
0.041
0.0033
0.36
0.23
1.05
1710
459.4
459.4
31.25
3
4.3
7
85
-71
4.6 X 109
2.6
729
IJRRAS 13 (3) ● December 2012
𝐶𝐶02 𝐻2
0
𝐶𝐶𝐻
3 𝐶𝑂𝑂𝐻
Adagiri & al. ● Effectiveness Factor for Porous Catalysts
[mole/m3]
[mole/m3]
16
10.5
6. RESULTS
This section presents the result of the developed model solution using orthogonal collocation numerical method with
third order semi-implicit Runge-Kutta method for the dimensionless concentration and temperature profiles and the
effectiveness factor as a function of Thiele modulus for the two studied reactions.
6.1Concentration and Temperature Profiles
The results of the dimensionless concentration profiles obtained from the Runge-Kutta solution of Equations 3.55
and 3.81 were obtained using FORTRAN programming. The plots of the concentration profiles are presented in
Figures 5.1, and 5.2 and that of temperature profiles are given in Figures 5.3 and 5.4 for the exothermic and
endothermic reactions, respectively.
6.2Effectiveness Factors
Modified Thiele moduli were obtained by using Equation (2.11). Effectiveness factors were obtained as functions of
modified Thiele modulus for varying γ and β in the two reactions. The parameter β and γ were obtained for selected
temperatures using Equations 2.17 and 2.20, respectively. Figures 5.5, 5.6, 5.7, 5.8, 5.9, 5.10, and 5.11 present the
exothermic effectiveness factors, while Figures 5.12, 5.13 and 5.14 present the endothermic reaction.
1.2
Dimensionless concentration
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Dimensionless radius
Figure 5.1: Dimensionless concentration profile for exothermic reaction
730
1.2
IJRRAS 13 (3) ● December 2012
www.arpapress.com/Volumes/Vol13Issue3/IJRRAS_13_3_08.pdf
530
528
Temperature (K)
526
524
522
520
518
516
0
0.2
0.4
0.6
0.8
Dimensionless radius
1
1.2
Figure 5.2: Exothermic reaction temperature profile
1
0.9
Dimensionless concentration
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
Dimensionless radius
Figure 5.3: Dimensionless concentration profile for endothermic reaction
731
1.2
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Adagiri & al. ● Effectiveness Factor for Porous Catalysts
480
478
476
Temperature (K)
474
472
470
468
466
464
462
0
0.2
0.4
0.6
0.8
Dimensionless radius
1
Figure 5.4: Endothermic reaction temperature profile
120
Effectiveness factor, η
100
80
60
40
20
0
0
0.1
0.2
0.3
0.4
0.5
Thiele modulus, Ø
γ =23.00 and β = 0.001933
Figure 5.5: Exothermic effectiveness factor for γ = 23.00 and β = 0.0187
732
0.6
IJRRAS 13 (3) ● December 2012
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Effectiveness factor, η
25
20
15
10
5
0
0
1
2
Thiele modulus, Ø
γ =20.13 and β…
3
Figure 5.6: Exothermic effectiveness factor for γ = 20.13 and β = 0.0163
8
Effectiveness factor, η
7
6
5
4
3
2
1
0
0
2
4
6
Thiele modulus, Ø
8
γ =17.13 and β = 0.001439
Figure 5.7: Exothermic effectiveness factor for γ = 17.13 and β = 0.0139
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2.5
Effectiveness factor, η
2
1.5
1
0.5
0
0
5
10
15
Thiele modulus, Ø
γ =16.10 and β = 0.001353
20
Figure 5.8: Exothermic effectiveness factor for γ = 15.25 and β = 0.0124
1.4
Effectiveness factor, η
1.2
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
Thiele modulus, Ø
γ=15.57 and β =0.001309
Figure 5.9: Exothermic effectiveness factor for γ = 14.38 and β = 0.0116
734
25
30
IJRRAS 13 (3) ● December 2012
Adagiri & al. ● Effectiveness Factor for Porous Catalysts
1.2
Effectiveness factor, η
1
0.8
0.6
0.4
0.2
0
0
10
20
Thiele modulus, Ø
30
40
γ =15.27 and β =0.001284
Figure 5.10: Exothermic effectiveness factor for γ = 13.65 and β = 0.0110
1.2
1
Effectiveness factor, η
0.8
0.6
0.4
0.2
0
0
10
20
30
40
Thiele modulus, Ø
γ =21.38 and β =-…
Figure 5.11: Endothermic effectiveness factor for γ = 21.38 and β = -0.00325
735
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IJRRAS 13 (3) ● December 2012
Adagiri & al. ● Effectiveness Factor for Porous Catalysts
1.2
Effectiveness factor, η
1
0.8
0.6
0.4
0.2
0
0
10
20
30
40
Thiele modulus, Ø
50
γ = 20.84 and β = -…
Figure 5.12: Endothermic effectiveness factor for γ = 20.84 and β = -0.00316
Effectiveness factor, η
1.2
1
0.8
0.6
0.4
0.2
0
0
20
40
60
80
Thiele modulus, Ø
γ = 19.58 and β =-…
Figure 5.13: Endothermic effectiveness factor for γ = 19.58 and β = -0.00298
736
100
IJRRAS 13 (3) ● December 2012
Adagiri & al. ● Effectiveness Factor for Porous Catalysts
Effectiveness factor, η
1.2
1
0.8
0.6
0.4
0.2
0
0
50
100
Thiele modulus, Ø
γ =18.85 and β =-…
150
Figure 5.14: Endothermic effectiveness factor for γ = 18.85 and β = -0.00287
7. DISCUSSION OF RESULTS
Figures 5.1 and 5.3 indicated that the concentration profiles of the reactants from the surface of the pellet to the
interior show decreasing trends for both exothermic and endothermic reactions, respectively. This would indicate
that as the reactants diffuse into the pores of the catalyst, reactions take place along the active sites located at the
pore walls, and when this is coupled with resistance posed by these walls to flow, the concentration is reduced. This
phenomenon is observed in both exothermic and endothermic catalytic heterogeneous reactions. Figure 5.2 shows
increasing temperature down from the surface to the interior of the catalyst for the exothermic reaction, while in
Figure 5.4 the reverse is the case for the endothermic reaction. For the exothermic reaction, heat is generated inside
the pellet and conducted to the surface fluid, while for the endothermic reaction, heat absolved by the pellet as
reaction occurs along the pore wall.
The effectiveness factors versus Thiele modulus for the exothermic reaction are shown in Figures 5.5 to 5.10 with β
> 0, and for the endothermic reaction in Figures 5.11 to 5.14 with β < 0 for all values of Ø. The profile shown in
Figure 5.9 was generated using values of β and γ calculated from the experimental data (T = 517K). Thus, at the
conditions specified in the experiment, effectiveness factors peak value were found to be slightly more than unity
(about 1.3) inside the pellet for the exothermic reaction. As the temperature was increased to 527K, values of
effectiveness factor were found to be less than unity throughout (Figure 5.10). Reductions in surface temperature
were however yielding correspondingly higher peak values which were much more than unity inside the exothermic
pellet (Figures 5.5 through 5.8). More significantly, in Figure 5.5, the maximum of the effectiveness factor was
calculated to be about 98 where the Thiele modulus was about 0.06. This compares with the value of 100 at a Thiele
modulus of 0.1 as reported by Carberry [9]. We need to consider under what circumstances is this high value of the
effectiveness factor possible.
To obtain the effect of more than normal value of the effectiveness factor, the pellet surface temperatures were
reduced in the model, thus giving higher values of β and γ. Are these lower values feasible for the exothermic
reaction? When the surface temperature was increased beyond the experimental temperature (517K), the
effectiveness profile was less than unity. However, when the surface temperature was reduced to 500K used in
Figure 5.8 where a maximum effectiveness factor of about 2.0 was calculated. Also, the temperature was reduced to
470K and 400K to get a maximum of about 7.6 and 24 in Figures 5.7 and 5.8 respectively; a further reduction to
350K resulted in a maximum of about 98 (Figure 5.5). That is, for the exothermic reaction, the lower the surface
737
IJRRAS 13 (3) ● December 2012
Adagiri & al. ● Effectiveness Factor for Porous Catalysts
temperature the higher the values of β and γ, the higher the peak values of the effectiveness factor. This implies that
we can get unreasonably high values of the effectiveness factor if the surface temperature is depressed sufficiently
enough. More importantly, for exothermic reactions, the unreasonably high values of the effectiveness factor
reported in the theoretically derived profiles are not useful, because the overall pellet activity beyond a temperature
for realistic reaction rate.
The phenomenon could not be obtained in the endothermic case in spite of the varying values of β and γ. This could
be explained looking back at the concentration and temperature profiles Figures 5.3 and 5.4, respectively. Although
the concentration of the reactant in either case drop from the pellet surface to the interior, the temperature of the
exothermic pellet and the reaction rate increases from the surface to the interior. That is, the dual effect of increasing
temperature and decreasing concentration for the exothermic pellet surface as we move to the interior of the pellet
account for this effect.
However, the endothermic pellet has both the pellet temperature and reactant concentration decreasing from the
pellet surface to the interior. Thus, at no point inside the endothermic pellet is the combined effect of both
temperature and rate surpass or even equal the reaction rate at surface conditions. This is, the reason the values of
the effectiveness factor cannot be higher than unity for all values Ø in the endothermic model is because these two
effects are in same direction.
8. CONCLUSION
The model developed predicted the effectiveness factor of Langmuir-Hinshelwood rate form for real exothermic and
endothermic reactions as functions of Thiele modulus, Ø, Arrhenius number, γ, and heat of reaction parameter, β,
satisfactorily through the specification of concentration and the temperature profiles in the pellet. Due to the
conflicting effect of temperature and concentration gradients on exothermic reaction rate, the exothermic
effectiveness factor can be larger than unity for certain, Ø, β, and γ. The magnitude of the peak value was increasing
with decreasing pellet temperature. The effectiveness factors for the endothermic reaction were all not larger unity
because the two gradients (temperature and concentration) reduces reaction rate from the surface to the interior of
the pellet. There were no significant differences in the profiles of the endothermic curves for different the surface
temperatures considered.
9. NOMENCLATURE
Aj, k
Orthogonal collocation matrix representing first derivative
An0
Zeroth Aris number
An1
First Aris Number
Bj, k
Orthogonal collocation matrix representing second derivative
CA
Concentration of key component A
Cpf
Specific heat capacity
𝑐
Dimensionless concentration
De
Effective diffusion coefficient
Ea
Activation energy
N
Number of collocation points
Nu
Nulsset number
n
Power rate order
Rg
Ideal gas constant
R
Pellet external radius
r
radius
Sh
Sherwood number
T
Temperature
𝑇𝑠𝑖,𝑘
Temperature vector at collocation points
t
Time
X0
Characteristic dimension
Peh
heat Peclet number related to the particle diameter
Pem
mass Peclet number related to the particle diameter
Greek symbols
α1 α2 and α3
constant defined in Equations 3.27 – 3.29
738
Dimensionless
Dimensionless
Dimensionless
Dimensionless
mol/m3
J/(Kg.K)
Dimensionless
m2/s
J/mol.
Dimensionless
Dimensionless
Dimensionless
J/(kgmol.K)
m
m
Dimensionless
K
Dimensionless
s
m
Dimensionless
Dimensionless
IJRRAS 13 (3) ● December 2012
β
γ
δ
ε
η
λ
ρ
τ
Ø
Adagiri & al. ● Effectiveness Factor for Porous Catalysts
the heat of reaction parameter,
Arrhenius number
Dimensionless radii
Voidage
Effectiveness factor
Effective thermal conductivity
Density
Dimensionless time
Thiele modulus
Dimensionless
Dimensionless
Dimensionless
Dimensionless
Dimensionless
W/(m.K)
kg/m3
Dimensionless
Dimensionless
Subscripts and Superscripts
i
Element index
j
jth collocation point
k
Iteration index
p
pellet properties
s
pellet surface condition
10. REFERENCES
[1].
[2].
[3].
[4].
E.W. Thiele, Relation between catalytic activity and size of particle, Ind. Eng. Chem., 31, Issue 7,. 916-920 (1939).
R. Aris: The mathematical theory of diffusion and reaction in permeable catalyst I. Oxford: Clarendon Press (1975).
R. Aris: The mathematical theory of diffusion and reaction in permeable catalyst II. Oxford: Clarendon Press (1975).
R.J. Wijngaarden, A.E. Kronberg, K.R. Westerterp: Industrial catalysis: Optimizing catalyst and processes. Weinheim:
Wiley-VCH (1998).
[5].
G.W. Roberts, C.N. Satterfield, Effectiveness factors for reversible reactions, Ind. Eng. Chem. Fund,. 7, 664-667 (1968).
[6].
A.A. Susu: Chemical kinetics and heterogeneous catalysis. Lagos: CJC Publishers (Nigeria) Limited (1997).
[7].
J.B. Anderson., A criterion for isothermal behavior of a catalyst pellet, Chem. Eng. Sci., 18, 147- (1963).
[8].
H. Kubota, Y. Yamanaka, I.G. Dalla Lana, Effective diffusivity of multicomponent gaseous reaction systems.
Application to catalyst effectiveness factor, J. Chem. Eng. Jpn., 2, 71-75 (1969).
[9].
J.J. Carberry: Chemical and Catalytic Reaction Engineerin, New York: McGraw-Hill Book Company (1976).
[10]. J.W. Villadsen,, W.E. Stewart, Solution of Boundary-Value Problems by Orthogonal Collocation, Chem. Eng. Sci., 22,
148 –1501 (1967).
[11]. J.W. Villadsen: Selected approximation methods for chemical engineering Problems, Copenhagen, Denmark: Tekniske
Hojskole (1970).
[12]. V. Hlavacek, M. Kubicek, J. Caha, Qualitative analysis of behaviour of nonlinear parabolic equations II- Applications of
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