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IJRRAS 13 (3) ● December 2012 www.arpapress.com/Volumes/Vol13Issue3/IJRRAS_13_3_08.pdf

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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,*

1Nordbound Integrated Engineering Services Ltd., P.O. Box 3111, Ikorodu, Lagos, Nigeria

2Department of Chemical Engineering, University of Maiduguri, Borno State, Nigeria

3Department 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

IJRRAS 13 (3) ● December 2012 Adagiri & al. ● Effectiveness Factor for Porous Catalysts

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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


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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


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


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Ø = 𝑛 𝑅 𝑘


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


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

evaluated at the collocation points:

𝑦 𝑥𝑗 = 𝑎𝑖𝑦𝑖 𝑥𝑗 𝑁+2𝑖=1 , 𝑗 = 1, … . . , 𝑁 + 2 (2.27)

and solved for the coefficients in terms of the solution at the collocation points:

𝑎𝑖 = 𝑦𝑖 𝑥𝑗 −1𝑁+2

𝑖=1 𝑦 𝑥𝑗 , 𝑖 = 1, . . . . , 𝑁 + 2 (2.28)

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

𝑑𝑥 𝑥𝑗 = 𝑦𝑖 𝑥𝑘 −1𝑁+2

𝑖 ,𝑘=1 𝑦 𝑥𝑘 𝑑𝑦𝑖

𝑑𝑥 𝑥𝑗 , 𝑗 = 1, . . . . , 𝑁 + 2 (2.29)

or shortened to 𝑑𝑦

𝑑𝑥 𝑥𝑗 = 𝐴𝑗𝑘

𝑁+2𝑖 ,𝑘=1 𝑦 𝑥𝑘 (2.30)

Rearranging, we have

𝐴𝑗𝑘 = 𝑦𝑖 𝑥𝑘 −1𝑁+2𝑖=1

𝑑𝑦𝑖

𝑑𝑥 𝑥𝑗 (2.31)

Similar steps can be applied to the second derivative to obtain 𝑑2𝑦

𝑑𝑥2 𝑥𝑗 = 𝐵𝑗𝑘𝑁+2𝑖 ,𝑘=1 𝑦 𝑥𝑘 , (2.32)

𝐵𝑗𝑘 = 𝑦𝑖 𝑥𝑘 −1𝑁+2𝑖=1

𝑑2𝑦𝑖

𝑑𝑥2 𝑥𝑗 (2.33)

For the solution of the catalyst pellet problem, orthogonal collocation is applied at the interior points

𝐵𝑗 ,𝑖𝐶𝑖 = ∅2𝑅 𝐶𝑗 , 𝑗 = 1, … , 𝑁𝑁+1𝑖=1 (2.34)

and the boundary condition is solved for

𝐶𝑁+1 = 1 (2.35)


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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 ≡ 𝑅 𝑐 𝑥 𝑥𝑎−110 𝑑𝑥

𝑅 𝑐 1 𝑥𝑎−110 𝑑𝑥

= 𝑊𝑗 𝑅 𝑐𝑗 ,

𝑁+1𝑖=1

𝑊𝑗 𝑅 1 𝑁+1𝑖=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.

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 ∂cs

∂r

r (3.2)

𝑜𝑢𝑡𝑝𝑢𝑡 𝑎𝑡 𝑟 + 𝛿𝑟 = −4𝜋εp 𝔇r ∂cs

∂r

r+δr (3.3)

𝑅𝑎𝑡𝑒 𝑜𝑓 𝑎𝑑𝑠𝑜𝑟𝑝𝑡𝑖𝑜𝑛 𝑜𝑛 𝑐𝑎𝑡𝑎𝑙𝑦𝑠𝑡 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 = 4𝜋r2 ∂r∂q

∂t (3.4)

𝑅𝑎𝑡𝑒 𝑜𝑓 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑜𝑛 𝑠𝑜𝑙𝑖𝑑 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑜𝑣𝑒𝑟 𝑡𝑕𝑒 𝑡𝑖𝑚𝑒 𝑝𝑒𝑟𝑖𝑜𝑑 =

4𝜋r2εp ∂r∂cs

∂t (3.5)

𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑑𝑢𝑒 𝑡𝑜 𝑐𝑕𝑒𝑚𝑖𝑐𝑎𝑙 𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 = 4𝜋r2 ∂rηi Ri (ciT) (3.6)

Inserting Equation 3.2 to 3.6 into Equation 3.1 yields

−4𝜋εp 𝔇r ∂cs

∂r

r − 4𝜋εp 𝔇r

∂cs

∂r

r+δr− 4𝜋r2 ∂r

∂q

∂t + (4𝜋r2εp ∂r

∂cs

∂t

= 4𝜋r2 ∂rεp ∂cs

∂T (3.7)

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 𝑟2𝔇r

∂csi

∂r − 𝜌𝑝

𝜕𝑞

𝜕𝑡− ηi Ri (ciT) = εp

∂cs

∂t (3.8)

εp 𝔇r(∂2𝑐𝑠𝑖

∂𝑟2 +2

𝑟

𝜕𝐶𝑠

𝜕𝑟) − 𝜌𝑝

𝜕𝑞

𝜕𝑡− ηi Ri (ciT) = εp

∂Cs

∂t (3.9)

𝑟 + 𝛿𝑟

r


722

Assuming steady state, we have:

εp 𝔇r ∂2csi

∂𝑟2 +2

𝑟

𝜕𝐶𝑠

𝜕𝑟 − 𝜌𝑝

𝜕𝑞

𝜕𝑡= ηi Ri (ciT) (3.10)

Initial and boundary conditions are:

1) t=0; R>r>0; Cs=0 (3.11)

2) r≥R t>0 𝔇r𝜕𝐶𝑠

𝜕𝑟= 𝐾𝑠(𝐶𝑠𝑖 − 𝐶𝑓)𝑟>𝑅 (3.12)

3) ∂Cs

∂r

r=0= 0, t > 0 (3.13)

Introducing dimensionless variables

i. r2 = R

2 𝛿 (3.14)

ii. ∂r =R

2𝛿1

2 𝜕𝛿 (3.15)

iii. ∂r2 =R2

4𝛿𝜕𝛿2 (3.16)

iv. t =τ

Uf𝑍𝑇 (3.17)

v. ∂t =∂τ

Uf𝑍𝑇 (3.18)

vi. cs =csi

c0 (3.19)

vii. τ =tUf

ZT (3.20)

viii. ∂τ =∂tUf

ZT (3.21)

ix. Qi∗ =

q i∗

q0i∗ (3.22)

Introducing the dimensionless variables into Equation 3.10, we have:

ℰ𝑝𝔇𝑟 𝑐0 ∂2cs

𝑅2∂𝛿2

4𝛿

+2𝑐0 ∂cs

𝑅 ∂12

𝑅 ∂𝛿

2𝛿12

−𝑞0

𝑐0

𝜕𝑄∗

𝜕𝑐𝑠 𝑈𝑓𝑐0

𝑍𝑇

𝜕𝑐𝑠

𝜕𝑧 = 𝜂𝑅(𝑐, 𝑇) (3.23)

ℰ𝑝𝔇𝑟 4𝛿𝑐0 ∂2cs

𝑅2 ∂𝛿2 +

4𝑐0 ∂cs

𝑅2 ∂𝛿 −

𝑞0

𝑐0

𝜕𝑄∗

𝜕cs 𝑈𝑓𝑐0

𝑍𝑇

𝜕cs

𝜕𝑧 = 𝜂𝑅(𝑐, 𝑇) (3.24)

ℰ𝑝𝔇𝑟4𝑐0𝛿

𝑅2 ∂2cs

∂𝛿2 +1

𝛿

𝜕𝑐𝑠

𝜕𝛿 −

𝑞0

𝑐0

𝜕𝑄∗

𝜕cs 𝑈𝑓𝑐0

𝑍𝑇

𝜕cs

𝜕𝑧 = 𝜂𝑅(𝑐, 𝑇) (3.25)

Multiply both sides by 𝑅2𝑍𝑇

𝔇𝑟𝑐0𝑈𝑓

We have: ℰ𝑝 4𝑍𝑇𝛿

𝑈𝑓 ∂2cs

∂𝛿2 +1

𝛿

∂cs

∂𝛿 − 𝜌𝑝

𝑅2

𝔇𝑟

𝑞0∗

𝑐0

𝜕𝑄∗

𝜕cs

𝜕cs

𝜕τ =

𝑅2𝑍𝑇

𝔇𝑟𝑐0𝑈𝑓𝜂𝑅(𝑐, 𝑇) (3.26)

Thus, we have:

𝛼1 =ℰ𝑝 4𝑍𝑇𝛿

𝑈𝑓 (3.27)

𝛼2 = 𝜌𝑝𝑅2

𝔇𝑟

𝑞0∗

𝑐0

𝜕𝑄∗

𝜕cs (3.28)

𝛼3 =𝑅2𝑍𝑇

𝔇𝑟𝑐0𝑈𝑓 (3.29)

𝛼1 ∂2cs

∂𝛿2 +1

𝛿

∂cs

∂𝛿 − 𝛼2

𝜕cs

𝜕τ = 𝛼3𝜂𝑅(𝑐, 𝑇) (3.30)

Assume mass transfer resistance is negligible, we have:

𝛼1 ∂2cs

∂𝛿2 +1

𝛿

∂cs

∂𝛿 = 𝛼3𝜂𝑅(𝑐, 𝑇) (3.31)


(1) cs = 0; τ ≤ 0; 1 ≥ 𝛿 ≥ 0 (3.32)


723

(2) 4

Sh

∂cs

∂𝛿

𝛿=1= cf − cs 𝛿=1 (3.33)

(3) ∂cs

∂𝛿

𝛿=0= 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 (3.35)

𝑖𝑛𝑝𝑢𝑡 𝑎𝑡 𝑟 = −4𝜋𝑟2휀𝑝𝜆𝑠 𝜕𝑦

𝜕𝑥 𝑟 ` (3.36)

𝑜𝑢𝑡𝑝𝑢𝑡 𝑎𝑡 𝑟 + 𝛿𝑟 = −4𝜋𝑟2휀𝑝𝜆𝑠 𝜕𝑦

𝜕𝑥 𝑟+𝛿𝑟

(3.37)

𝑟𝑎𝑡𝑒 𝑜𝑓 𝑎𝑑𝑠𝑜𝑟𝑝𝑡𝑖𝑜𝑛 𝑜𝑛 𝑐𝑎𝑡𝑎𝑙𝑦𝑠𝑡 𝑖𝑛𝑛𝑒𝑟 𝑠𝑢𝑟𝑓𝑎𝑐𝑒

= 4𝜋𝑟2𝜕𝑟 Δ𝐻𝜌𝑝𝜕𝑞

𝜕𝑡 (3.38)

𝑟𝑎𝑡𝑒 𝑜𝑓 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑜𝑛 𝑠𝑜𝑙𝑖𝑑 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑜𝑣𝑒𝑟 𝑡𝑕𝑒 𝑡𝑖𝑚𝑒 𝑝𝑒𝑟𝑖𝑜𝑑

= 4𝜋𝑟2𝜕𝑟휀𝑝 𝜌𝑠 𝐶𝑝 ,𝑠 𝜕𝑇𝑠

𝜕𝑡 (3.39)

𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑑𝑢𝑒 𝑡𝑜 𝑐𝑕𝑒𝑚𝑖𝑐𝑎𝑙 𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 = 4𝜋𝑟2𝜕𝑟 Δ𝐻r ηiRi 𝑐𝑖 , 𝑇 (3.40)

Assuming steady state and negligible heat transfer resistances, we have

−4𝜋𝑟2휀𝑝𝜆𝑠 𝜕𝑦

𝜕𝑥 𝑟 — 4𝜋𝑟2휀𝑝𝜆𝑠

𝜕𝑦

𝜕𝑥 𝑟+𝛿𝑟

= 4𝜋𝑟2𝜕𝑟 Δ𝐻r ηiRi 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 4𝜋𝑟2𝜕𝑟, we have:

휀𝑝 𝜕

𝜕𝑡 𝑥2𝜆𝑠

𝜕𝑇𝑠

𝜕𝑟 = −Δ𝐻r ηiRi 𝑐𝑖 , 𝑇 (3.42)


i. 𝑡 ≤ 0, 𝑅 > 𝑟 > 0, 𝑇𝑠 = 𝑇𝑤 = 𝑇0 (3.43)

ii. 𝑟 = 𝑅, 𝑡 > 0; −𝜆𝑠 𝜕𝑇𝑠

𝜕𝑟= 𝛼𝑠 𝑇𝑠 − 𝑇𝑓 (3.44)

iii. 𝑟 = 0, 𝑡 > 0; 𝜕𝑇𝑠

𝜕𝑟 𝑟=0

= 0 (3.45)

Introducing the following dimensionless variables:

i. 𝑟2 = 𝑅2𝛿 (3.46)

ii. 𝜕𝑟 =𝑅

2𝛿1/2 𝜕𝛿 (3.47)

iii. 𝜕𝑟2 =𝑅

4𝛿 𝜕𝛿2 (3.48)

iv. 𝑡 =𝜏

𝑈𝑓𝑍𝑇 (3.49)

v. 𝜕𝑡 =𝜕𝜏

𝑈𝑓𝑍𝑇 (3.50)

vi. 𝑇𝑠 =

𝑇𝑠

𝑇0 (3.51)

vii. 𝜏 =𝑡𝑈𝑓

𝑍𝑇 (3.52)

viii. 𝜕𝜏 =𝜕𝑡𝑈𝑓

𝑍𝑇 (3.53)

ix. 𝑄1∗ =

𝑞∗

𝑞0 (3.54)

Introducing these dimensionless variables into Equation 3.42, we have: 4휀𝑝 𝜆𝑠

𝑅2 𝜕2𝑇𝑠

𝜕𝛿2 +1

𝛿

𝜕𝑇 𝑠

𝜕𝛿 =

1

𝑇0 −Δ𝐻𝑟 𝜂𝑖𝑅𝑖 𝑐𝑖 , 𝑇𝑖 (3.55)

Subject to initial and boundary conditions:

i. 𝑇 𝑠 = 1; 𝜏 ≤ 0; 1 ≥ (3.56)

ii. 4

𝑁𝑢 𝜕𝑇 𝑠

𝜕𝛿 𝛿=1

= 𝑇 𝑓 − 𝑇 𝑠 𝛿=1 (3.57)

iii. 𝜕𝑇 𝑠

𝜕𝛿 𝛿=0

= 0; 𝜏 > 0 (3.58)

Let 𝛼4 =4휀𝑝 𝜆𝑠

𝑅2


724

𝛼4 𝜕2𝑇𝑠

𝜕𝛿2 +1

𝛿

𝜕𝑇

𝜕𝛿 =

1


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: ∂cs

∂Z = 𝐴𝑗 ,𝑘cs 𝑖 ,𝑘

𝑁+1𝐾=1 (4.1)

∂2cs

∂Z 2 = 𝐵𝑗 ,𝑘cs 𝑖 ,𝑘𝑁+1𝐾=1 (4.2)

Substituting Equations 4.1 and 4.2 into Equation 3.31, we have:

𝛼1 𝐵𝑗 ,𝑘cs 𝑖 ,𝑘𝑁+1𝐾=1 +

1

𝛿𝑗 𝐴𝑗 ,𝑘cs 𝑖 ,𝑘

𝑁+1𝐾=1 = 𝛼3𝜂𝑅(𝑐, 𝑇) (4.3)

Expanding Equation 4.3:

𝛼1 𝐵𝑗 ,𝑘cs 𝑖 ,𝑘𝑁+1𝐾=1 + 𝐵𝑗 ,𝑘cs 𝑖 ,𝑁+1

+1

𝛿𝑗 𝐴𝑗 ,𝑘cs 𝑖 ,𝑘

𝑁+1𝐾=1 + 𝐴𝑗 ,𝑘cs 𝑖 ,𝑁+1

= 𝛼3𝜂𝑅(𝑐, 𝑇) (4.4)

Factorizing like terms, we have:

𝛼1 𝐵𝑗 ,𝑘 +𝑁𝐾=1

1

𝛿𝑗 𝐴𝑗 ,𝑘

𝑁+1𝐾=1 cs 𝑖 ,𝑘

+ 𝐵𝑗 ,𝑁+1 +1

𝛿𝑗𝐴𝑗 ,𝑁+1 cs 𝑖 ,𝑁+1

= 𝛼3𝜂𝑅(𝑐, 𝑇) (4.5)

To substitute for cs 𝑖 ,𝑁+1, the concentration at the surface of the pellets, we use Equation 3.33:

4

Sh 𝐴𝑁+1,𝑘 +𝑁+1

𝐾=1 cs 𝑖 ,𝑘 = cf 𝑖 ,𝑗

− cs 𝑖 ,𝑁+1 (4.6)

Therefore,

cs 𝑖 ,𝑁+1= Φ1cf 𝑖 ,𝑗

− 𝐴𝑁+1,𝑘 +𝑁+1𝐾=1 cs 𝑖 ,𝑘

(4.7)

where

Φ1 =1

1+4

Sh𝐴𝑁+1, 𝑁+1

(4.8)

𝛼1 𝐵𝑗 ,𝑘 +𝑁𝐾=1

1


𝑁𝐾=1 cs 𝑖 ,𝑘

+ 𝐵𝑗 ,𝑁+1 +1

𝛿𝑗𝐴𝑗 ,𝑁+1 Φ1cf 𝑖 ,𝑗

− 𝐴𝑁+1,𝑘 +𝑁+1𝐾=1 cs 𝑖 ,𝑘

= 𝛼3𝜂𝑅(𝑐, 𝑇)

(4.9)

Thus we have,

𝛼1 𝐵𝑗 ,𝑘 +

𝑁

𝐾=1

1

𝛿𝑗

𝐴𝑗 ,𝑘 −

𝑁

𝐾=1

𝐵𝑗 ,𝑁+1𝐴𝑁+1,𝑘′ −

1

𝛿𝑗

𝐴𝑗 ,𝑁+1𝐴𝑁+1,𝑘′ cs 𝑖 ,𝑘

+ 𝐵𝑗 ,𝑁+1Φ1 +1

𝛿𝑗

𝐴𝑗 ,𝑁+1Φ1 cs 𝑖 ,𝑘

= 𝛼3𝜂𝑅(𝑐, 𝑇) (4.10)

Therefore,

𝐹 𝑗 = 𝛼1 𝐵𝑗 ,𝑘 +𝑁𝐾=1

1

𝛿𝑗 𝐴𝑗 ,𝑘 −𝑁

𝐾=1 𝐵𝑗 ,𝑁+1𝐴𝑁+1,𝑘′ −

1

𝛿𝑗𝐴𝑗 ,𝑁+1𝐴𝑁+1,𝑘

′ cs 𝑖 ,𝑘+ 𝐵𝑗 ,𝑁+1Φ1 +

1

𝛿𝑗𝐴𝑗 ,𝑁+1Φ1 cf 𝑖 ,𝑗

−

𝛼3𝜂𝑅(𝑐, 𝑇) (4.11)

4.2Application of Orthogonal Numerical Technique on Heat Balance Equation

Using orthogonal numerical technique as in mass balance equation, we have:

𝛼4 𝐵𝑗 ,𝑘𝑇 𝑠𝑖 ,𝑘𝑁+1𝑘=1 +

1

𝛿𝑗 𝐴𝑗 ,𝑘𝑇 𝑠𝑖 ,𝑘

𝑁+1𝑘=1 =

1


Expanding Equation 4.12, we have:

𝛼4 𝐵𝑗 ,𝑘𝑇 𝑠𝑖 ,𝑘𝑁+1𝑘=1 + 𝐵𝑗 𝑁+1

𝑇 𝑠𝑁+1 +

1

𝛿𝑗 𝐴𝑗 ,𝑘𝑇 𝑠𝑖 ,𝑘

+ 𝐴𝑗 𝑁+1𝑇 𝑖 ,𝑁+1

𝑁𝑘=1 =

1

𝑇0 −Δ𝐻𝑟 𝜂𝑖𝑅𝑖 𝑐𝑖 , 𝑇𝑖

(4.13)

Factorizing like terms we have:

𝛼4 𝐵𝑗 ,𝑘𝑁𝑘=1 +

1


𝑁𝑘=1 𝑇 𝑠𝑖 ,𝑘

+ 𝐵𝑗 𝑁+1+

1

𝛿𝑗𝐴𝑗 ,𝑁+1 𝑇 𝑠𝑖 ,𝑁+1

=1


Substituting for 𝑇𝑠𝑖 ,𝑁+1, using Equation 3.57, we have:


725

4

𝑁𝑢 𝐴𝑁+1𝑇𝑠𝑖 ,𝑘

𝑁+1𝑘=1 = 𝑇 𝑓 𝑖 ,𝑗

− 𝑇𝑠𝑖 ,𝑁+1 (4.15)

Thus, we have:

𝑇 𝑠𝑖 ,𝑁+1= 𝛷2𝑇 𝑖 ,𝑗 − AN+1,k

′′ T s i,kN+1k=1 (4.16)

Therefore,

𝑇 𝑠𝑖 ,𝑁+1= 𝛷2𝑇 𝑓 𝑖 ,𝑗

− 𝐴𝑁+1,𝑘′′ 𝑇𝑠𝑖 ,𝑘

𝑁+1𝑘=1 (4.17)

where

𝛷2 =1

1+4

𝑁𝑢𝐴𝑁+1,𝑁+1

(4.18)

𝛼1 𝐵𝑗 ,𝑘𝑁𝑘=1 +

1


𝑁𝑘=1 𝑇 𝑠𝑖 ,𝑘

+ 𝐵𝑗 𝑁+1+

1

𝛿𝑗𝐴𝑗 ,𝑁+1 𝛷2𝑇𝑓 𝑖 ,𝑗

− 𝐴𝑁+1,𝑘′′ 𝑇 𝑠𝑖 ,𝑘

1𝑘=1

= −𝛥𝐻𝑟 𝜂𝑖𝑅𝑖 𝑐𝑖 , 𝑇𝑖 (4.19)

Thus we have,

𝛼4

𝐵𝑗 ,𝑘𝑁𝑘=1 +

1


𝑁𝑘=1 − 𝐵𝑗 𝑁+1

𝐴𝑁+1,𝑘′′ −

1


′′ 𝑇𝑠𝑖 ,𝑘

+ 𝐵𝑗 ,𝑁+1𝛷2 +1

𝛿𝑗𝐴𝑗 ,𝑁+1𝛷2 𝑇𝑓 𝑖 ,𝑗

= −𝛥𝐻𝑟 𝜂𝑖𝑅𝑖 𝑐𝑖 , 𝑇𝑖

(4.20)

Therefore we have,

𝐹𝑗 = 𝛼1 𝐵𝑗 ,𝑘 +𝑁𝑘=1

1


𝑁𝑘=1 − 𝐵𝑗 𝑁+1

𝐴𝑁+1,𝑘′′ −

1


′′ 𝑇𝑓 𝑖 ,𝑘 — 𝛥𝐻𝑟𝜂𝑖𝑅𝑖 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

subroutines. JCOBI SUBROUTINE calculates the zeros 𝑃𝑤𝛼 ,𝛽 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 single-

step 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.


726

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


727

Figure 4.1: Dimensionless concentration and temperature flow chart

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

X


728

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.

𝐶𝐻2𝑂 + 1

2𝑂2

𝑘1 𝐶𝑂 + 𝐻2𝑂 (5.1)

−𝑟1 =𝑘1𝐶𝐶𝐻2𝑂

0.5

1+0.2𝐶𝐶𝐻2𝑂0.5 (5.2)

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


729

𝑘1 = 5.4 X 105exp(−66944

𝑅𝑔𝑇) (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

𝐶𝑂20 [mole/m

3] 34

𝐶𝐶𝐻2𝑂 0 [mole/m

3] 1.74

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:

𝐶2𝐻2 + 𝐶𝐻3𝐶𝑂𝑂𝐻 𝑅 𝐶𝐻3𝐶𝑂2𝐶𝐻𝐶𝐻2 (5.4)

𝑅 =𝑘∞ exp (−𝐸/𝑅𝑔𝑇)𝑃𝐶2𝐻2

1+exp −∆𝐻1𝑅𝑔𝑇

exp ∆𝑆1𝑅𝑔

𝑃𝐶𝐻3𝐶𝑂𝑂𝐻 +𝐾1𝑝𝐶𝐻3𝐶𝑂2𝐶𝐻𝐶 𝐻2

(5.5)

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 [m] 1

dt [m] 0.041

dp [m] 0.0033

Ε [ ] 0.36

us [m/s] 0.23

ρf [kg/m3] 1.05

cpf [J/(kg.K)] 1710

Tin [K] 459.4

Tw [K] 459.4

ΔH [J/mol] 31.25

Pehr [ ] 3

Pemr [ ] 4.3

Bi [ ] 7

Ea [kJ/(mole)] 85

ΔS1 [J/mole.K] -71

k∞ [mole/m3 cat s atm-1] 4.6 X 109

K1 [atm-1] 2.6


730

𝐶𝐶2𝐻2

0 [mole/m3] 16

𝐶𝐶𝐻3𝐶𝑂𝑂𝐻0 [mole/m3] 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.

Figure 5.1: Dimensionless concentration profile for exothermic reaction

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

Dim

en

sio

nle

ss

co

nc

en

tra

tio

n

Dimensionless radius

IJRRAS 13 (3) ● December 2012 www.arpapress.com/Volumes/Vol13Issue3/IJRRAS_13_3_08.pdf

731

Figure 5.2: Exothermic reaction temperature profile

Figure 5.3: Dimensionless concentration profile for endothermic reaction

516

518

520

522

524

526

528

530

0 0.2 0.4 0.6 0.8 1 1.2

Te

mp

era

ture

(K

)


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2

Dim

en

sio

nle

ss c

on

cen

trati

on



732

Figure 5.4: Endothermic reaction temperature profile

Figure 5.5: Exothermic effectiveness factor for γ = 23.00 and β = 0.0187

462

464

466

468

470

472

474

476

478

480

0 0.2 0.4 0.6 0.8 1

Te

mp

era

ture

(K

)


0

20

40

60

80

100

120

0 0.1 0.2 0.3 0.4 0.5 0.6

Eff

ec

tive

ne

ss

fa

cto

r, η

Thiele modulus, Ø

γ =23.00 and β = 0.001933


733



0

5

10

15

20

25

0 1 2 3

Eff

ecti

ve

ne

ss

fa

cto

r, η

Thiele modulus, Ø

γ =20.13 and β …

0

1

2

3

4

5

6

7

8

0 2 4 6 8 10

Eff

ecti

ve

ne

ss

fa

cto

r, η

Thiele modulus, Ø

γ =17.13 and β = 0.001439


734



0

0.5

1

1.5

2

2.5

0 5 10 15 20

Eff

ecti

ven

ess

fact

or,

η

Thiele modulus, Ø

γ =16.10 and β = 0.001353

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20 25 30

Eff

ec

tive

ne

ss

fa

cto

r, η

Thiele modulus, Ø

γ=15.57 and β =0.001309


735


Figure 5.11: Endothermic effectiveness factor for γ = 21.38 and β = -0.00325

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40

Eff

ec

tive

ne

ss

fa

cto

r, η

Thiele modulus, Ø

γ =15.27 and β =0.001284

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50Eff

ec

tive

ne

ss

fa

cto

r, η

Thiele modulus, Ø

γ =21.38 and β =-…


736



0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50

Eff

ec

tive

ne

ss

fa

cto

r, η

Thiele modulus, Ø

γ = 20.84 and β = -…

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60 80 100

Eff

ec

tive

ne

ss

fa

cto

r, η

Thiele modulus, Ø

γ = 19.58 and β =-…


737


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

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150

Eff

ec

tive

ne

ss

fa

cto

r, η

Thiele modulus, Ø

γ =18.85 and β =-…


738

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 Dimensionless

An0 Zeroth Aris number Dimensionless

An1 First Aris Number Dimensionless

Bj, k Orthogonal collocation matrix representing second derivative Dimensionless

CA Concentration of key component A mol/m3

Cpf Specific heat capacity J/(Kg.K)

𝑐 Dimensionless concentration Dimensionless

De Effective diffusion coefficient m2/s

Ea Activation energy J/mol.

N Number of collocation points Dimensionless

Nu Nulsset number Dimensionless

n Power rate order Dimensionless

Rg Ideal gas constant J/(kgmol.K)

R Pellet external radius m

r radius m

Sh Sherwood number Dimensionless

T Temperature K

𝑇 𝑠𝑖 ,𝑘 Temperature vector at collocation points Dimensionless

t Time s

X0 Characteristic dimension m

Peh heat Peclet number related to the particle diameter Dimensionless

Pem mass Peclet number related to the particle diameter Dimensionless

Greek symbols

α1 α2 and α3 constant defined in Equations 3.27 – 3.29


739

β the heat of reaction parameter, Dimensionless

γ Arrhenius number Dimensionless

δ Dimensionless radii Dimensionless

ε Voidage Dimensionless

η Effectiveness factor Dimensionless

λ Effective thermal conductivity W/(m.K)

ρ Density kg/m3

τ Dimensionless time Dimensionless

Ø Thiele modulus Dimensionless

Subscripts and Superscripts

i Element index

j jth collocation point

k Iteration index

p pellet properties

s pellet surface condition

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[2]. R. Aris: The mathematical theory of diffusion and reaction in permeable catalyst I. Oxford: Clarendon Press (1975).

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[4]. R.J. Wijngaarden, A.E. Kronberg, K.R. Westerterp: Industrial catalysis: Optimizing catalyst and processes. Weinheim:

Wiley-VCH (1998).

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[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

the method for estimation of domains of multiplicity, Chem. Eng. Sci., 26, 1743- 1752 (1971).

[13]. L.C. Windes, M.J. Schwedock, W.H. Ray, Steady state and dynamic modeling of a packed bed reactor for the partial

oxidation of methanol to formaldehyde I. Chem. Eng. Comm. 78, 1-7 (1989).

[14]. J.M. Valstar P.J. Van Den Berg, J. Ouserman, Comparison between two dimensional fixed bed reactor calculations and

measurements, Chem. Eng. Sci., 30, 723-728 (1974).

[15]. R. Aris., On shape factors for irregular particles, Chem. Eng. Sci.. 6, 262-268 (1957).

[16]. E.E. Petersen: Chemical Reaction Analysis, Englewoods Cliffs, New Jersey: Prentice-Hall New Jersey (1965).

[17]. R.A. Rajadhyaksha, K. Vaduseva, A simplified representation for nonisothermal effectiveness factor, J. Catal., 34, 321-

323 (1974).

[18]. K.B. Bischoff, Asymptotic solutions for gas absorption and reaction, Chem. Eng. Sci., 29, 1348-1349 (1974).

[19]. B. Valdman, R. Hughes, A simple method of calculating effectiveness factor for heterogeneous catalytic gas-solid

reactions, AIChE J., 30, 723-728 (1976).

[20]. J.L. Ibanez, Stability analysis of steady states in a catalytic pellet, J. Chem. Phys., 71, 5253-5256 (1979).

[21]. A. Namjoshi, B.D. Kulkarni, and L.K. Doraiswamy, A simple method of solution for a class of reaction-diffusion

problems, AIChE J., 29, 521-523 (1983).

[22]. L.K. Doraiswamy, M.M. Sharma: Heterogeneous reactions: Analysis, examples, and reactor design, vol.1. New York:

John Wiley and Sons (1983).

[23]. D.E. Mears, Tests for transport limitation in experimental catalytic reactors, Ind. Eng. Chem. Prod. Res. Dev., 10, 541-

547 (1971).

[24]. P.B. Weisz and Hicks, J. S., The behavior of porous catalytic particles in view of internal mass and heat diffusion effects.

Chem. Eng. Sci., 17, 265-275 (1962).

[25]. J.M. Thomas, W.J. Thomas: Principle and practice of heterogeneous catalysis. Weinheim: Wiley-VCH (1997)

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