solid catalyzed reactions

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Process Engineering Guide: GBHE-PEG-RXT-808

Solid Catalyzed Reactions

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CONTENTS Page 0 INTRODUCTION/PURPOSE 3 1 SCOPE 3 2 FIELD OF APPLICATION 3 3 DEFINITIONS 3 4 GENERAL BACKGROUND 4

4.1 General Considerations 4

5 SOLID CATALYZED GAS REACTIONS 7

5.1 Reaction Kinetics 7 5.2 Tests for Transport Limitations 13 5.3 Building a Reaction Kinetic Equation 15



6 INTRAPARTICLE 19

6.1 Types of Pore System 19 6.2 The Catalyst Effectiveness Factor 20 6.3 The Measurement of Effective Diffusivity 25

7 ENHANCEMENT OF INTRAPARTICLE 30 8 NOMENCLATURE 33

8.1 Dimensionless Parameters 34 8.2 Greek Letters 34 8.3 Subscripts 35

9 BIBLIOGRAPHY 35 9.1 Further Reading 36



APPENDICES A LANGMUIR - HINSHELWOOD KINETICS 37 FIGURES 1 EFFECTIVE RATE CONSTANT 7 2 ITERATIVE APPROACH TO REACTOR MODEL

DEVELOPMENT 8 3 COMMON LABORATORY MICROREACTORS (FLOW TYPE) 9 4 THE BERTY REACTOR 11 5 STEPS IN BUILDING A REACTION RATE EQUATION 15 6 A CENTRAL-COMPOSITE DESIGN FOR TWO FACTORS 17 7 FIRST ORDER ISOTHERMAL IRREVERSIBLE

REACTION WITHIN A CATALYST SPHERE 21 8 INTEGRAL YIELD vs CONVERSION SHOWING EFFECT OF

PELLET DIFFUSION 24 9 PREDICTED AND EXPERIMENTAL EFFECTIVENESS

FACTORS 27 10 STRUCTURAL PERMEABILITY vs PRESSURE PARAMETER Z

FOR BI-MODAL SUPPORTS 29 11 EFFECTIVENESS FACTOR vs THIELE MODULUS AND

INTRAPARTICLE PECLET NUMBER 31 12 RELATIVE INCREASE IN CATALYST PERFORMANCE 32



TABLES 1 EXAMPLES OF REACTOR TYPES 4 2 TYPICAL OPERATING PARAMETERS 5 3 BSERVED REACTION KINETIC PARAMETERS 6 4 REACTOR RATINGS 12 5 EXPERIMENTAL TESTS FOR INTRAPARTICLE AND

INTERPHASE GRADIENTS 14

6 HEURISTICS FOR BUILDING A HYPERBOLIC-TYPE RATE EQUATION 19

7 TYPES OF PORE SYSTEM 20 8 COMPARISON OF PORE DIFFUSIVITIES BETWEEN THE

DIFFUSION CELL AND PACKED BED METHODS 26



0 INTRODUCTION / PURPOSE This Guide is one of a series produced under the auspices of the GBH Enterprises.

1 SCOPE

This Guide covers the reaction kinetics and diffusional aspects of solid catalyzed reactions. It does not deal with either the fluid dynamics or the mechanical design of reactors for solid Catalyzed reactions.

2 FIELD OF APPLICATION

This Guide applies to process engineers in GBH Enterprises world-wide.

3 DEFINITIONS

For the purposes of this Guide, the following definitions apply: Effectiveness Factor The ratio of the global rate to the intrinsic rate of

reaction. Thiele Modulus A dimensionless group representing the relative

importance of diffusion and reaction taking place in a pellet.

With the exception of terms used as proper nouns or titles, those terms with initial capital letters which appear in this document and are not defined above are defined in the Glossary of Engineering Terms.



4 GENERAL BACKGROUND 4.1 General Considerations There are numerous possibilities for technical scale realization of solid catalyst reactors. This is illustrated in the following Table 1 which gives examples.



TABLE 1 EXAMPLES OF REACTOR TYPES

Further information about currently operated reactors may be found by retrieving information from a data base, see GBHE-PEG-RXT-800, Clause 9.



Although no type is precluded from featuring in a batch process, the boiling bed is the most usual one to use as a batch reactor. . Some general pointers to reactor choice and operating conditions are given in Table 2. TABLE 2 TYPICAL OPERATING PARAMETERS

Mixing: 5(i) and 5(ii) demand good premixing. If also 2(ii), might use

back-mixed fluidized bed. 5(iii) and 1(i) call for plug flow.

Temperature: 1(i) and 1(ii) likely to have an optimum temperature. 2(i) with 3(i) needs interbed cooling, cold shot or tubular. 2(i) with 3(iii) needs interbed heating, hot shot, or tubular.

Pressure: 2(i) with 4(i) suggests low pressure.

2(i) with 4(iii) suggests high pressure.

Catalyst: 5(i) surface coating, 5(ii) low porosity particles, surface impregnated, 5(iii) large particles, high internal surface area.

The classification of chemical reaction rate as more or less fast compares chemical rate with the rate of diffusive steps. The chemical rate is said to be very fast relative to the rate at which reacting species can penetrate from the bulk fluid to the catalyst particle. The rate per unit volume of particle will asymptote to the first order expression.



where:

k = mass transfer coefficient L/t A = catalyst particle bulk surface area L2 V = catalyst particle bulk volume L3 C = reactant concentration moles/L3

The chemical rate is described as fast when it is fast relative to the catalyst particle diffusive processes, but slow relative to external mass transfer. Here, the rate per unit volume of particle will asymptote to

where: k R = reaction rate constant t-1 (moles/L3) -(m-1) m = order of reaction ƞ = effectiveness factor

for ɸ 10, isothermal conditions.

ɸ = Thiele Modulus

where: D = effective diffusivity L2/t

For a fuller discussion of these aspects, see Satterfield [Ref. 2].



A computer program is available for calculating effectiveness factors for complex reactions in non-isothermal conditions, with a rigorous treatment of multi-component diffusion. See GBHE-PEG-RXT-819. If the chemical rate is slow relative to the inner diffusive steps; then the reaction rate per unit volume of particle will asymptote to :

i.e. ƞ = 1 The observed reaction order (m') and the activation energy (Eapp) depend upon the particular rate - controlling regime, as indicated in Table 3 below: TABLE 3 OBSERVED REACTION KINETIC PARAMETERS

Figure 1 illustrates the full range for a spherical particle with first order reaction. K eff = effective first order reaction rate constant.



FIGURE 1 EFFECTIVE RATE CONSTANT

5 SOLID CATALYZED GAS REACTIONS

This section reviews progress and suggests guidelines in the areas of reaction kinetics and intrapellet diffusion. The ideal of a mechanistic modeling approach is unlikely to be reached by the kind of research program feasible in an industrial laboratory. Limited resources, and particularly time, make it essential to identify a compromise between this ideal and a completely empirical approach. However, the precise synthesis of chemical transformations, occurring at the catalyst surface, and the restricted rates of mass and heat transfer to and from the surface, into the form of a practical mathematical model, containing the smallest number of independently measurable parameters, is far from obvious. A detailed case study is provided in GBHE-PEG-RXT-815 (The Selective Oxidation of n-Butane to Maleic Anhydride) to illustrate the philosophy. 5.1 Reaction Kinetics 5.1.1 Objectives

A sensible approach would demand the writing down of the bare minimum number of reactions necessary to describe key features of the observed product distribution, such as conversion and selectivity to desired product, and key byproducts which must be removed downstream from the reactor.



Then, classical rate equations of either the power-law type e.g.

or of the Langmuir-Hinshelwood form (hyperbolic)(see Appendix A and [Refs. 1 and 2]), for example:

are written for each reaction.

The model parameters k1, α and β or k2, KA and KB are estimated by fitting the proposed model to product distribution data. (see GBHE-PEG-RXT-821) Preferably, isothermal rate data should be gathered in the chemical rate-controlling regime. Alternatively, apparent kinetics can be developed on the actual-sized catalyst particles to be used in the reactor, although this removes one important variable - particle size – from subsequent optimization. These data, together with the residence time distribution (RTD) and heat and mass transfer data (or correlations) can be assembled into a reactor model which offers the prospect of modest (but certainly not wide-ranging) extrapolations and practical assessment of alternative reactor designs. Such an approach can be achieved with a modest outlay of resources and in a reasonable time scale of months, rather than years. It will usually be necessary to test the model against other data from which it was not derived (e.g. semi-technical or pilot-plant) in order to build confidence in its utility. Iteration back and forth between laboratory and pilot plant is inevitable. The scheme of model building is portrayed in Figure 2.



FIGURE 2 ITERATIVE APPROACH TO REACTOR MODEL DEVELOPMENT

5.1.2 Laboratory Reactors

In choosing a laboratory reactor for kinetic studies the following points need to be considered:

(a) ease of accurate chemical analysis.

(b) insurance of isothermality;

(c) an ideal RTD (i.e. plug flow or CSTR);

(d) absence of inter-phase concentration and temperature gradients;

(e) absence of intraparticle concentration and temperature gradients;

(f) ease of data analysis;

(g) cost and ease of construction;

(h) Ability to study deactivating catalysts.



Unfortunately, no single reactor simultaneously meets all these criteria. The preferred approach, if convenient, is to employ two types of laboratory reactor which, when combined, span the range of requirements. Laboratory reactors are classified under three types - differential, integral or CSTR (recycle), as depicted in Figure 3. FIGURE 3 COMMON LABORATORY MICROREACTORS (FLOW TYPE)

5.1.2.1 Differential Type

A differential reactor comprises typically a short section of 1/8" or 1/4" stainless steel tubing containing a small quantity of powdered catalyst (<1 gm) supported on quartz wool and held in a temperature controlled oven or furnace. The temperature within the catalyst bed is measured with the aid of a fine thermocouple and sample points are fitted fore and aft the bed in order to measure the small composition changes across the reactor (∆CA 0.1Co, typically). From the measured conversion, mass of catalyst (W) and reactant flow rate (FA), the reaction rate may be estimated directly:-



Because the conversion is small, isothermality can usually be assured and the RTD is unimportant, although by-passing is a potential hazard.

The greatest practical limitation of the differential reactor is the intrinsic difficulty of determining ∆CA. Rather than relying on the accurate measurement of small changes in reactant composition, it may actually be preferred to estimate ∆CA from product composition measurements and stoichiometry.

It is obviously necessary to produce synthetic feeds covering the ranges of reactant and product compositions anticipated within the commercial reactor. This aspect, too, may prove troublesome.

5.1.2.2 Integral Type

Integral reactors are also of the tubular type and their dimensions may vary widely from the microreactor scale (0.15m x 0.006m diameter packed with crushed catalyst) to a single plant scale reactor tube (5m x 0.028m diameter, say, containing 5 mm catalyst extrudates). High conversions, typical of commercial practice, are sought over this type of reactor, perhaps as high as 95% in certain cases. Now, of course, it is much more difficult to ensure isothermality of the bed, and special cooling (or heating) arrangements are often necessary, such as immersing the reactor in a molten salt bath or a fluidized bed. It may even be necessary to dilute the catalyst with inert packing in order to limit the rate of reaction per unit bed volume.

Because these reactors tend to be long and narrow, the ideal of plug flow is closely approached. However, since the reactant conversion changes considerably over the reactor, Equation (1) must be replaced by the differential form :



It now becomes necessary to infer a functional form for r (CA), insert it into Equation (2) and integrate to obtain CA for comparison with observations of CA at various values of W/FA. This is obviously more complicated than using Equation (1) to determine the reaction rate directly.

A comparison of reaction kinetics for a complex commercial reaction system – xylenes isomerization - using both differential and integral reactors is given by Orr and co-workers [Ref.!3].

High temperature "shift" kinetics employing integral reactors are dealt with in some depth by Chinchen and Spencer [Ref. 4].

5.1.2.3 CSTR Type

Bearing in mind the limitations of the differential and integral reactors mentioned above, much effort has gone into developing the so-called "ideal" gradientless reactor of the CSTR type. The preferred variant seems to be the internal recirculation or "Berty" reactor, as it has come to be known. One such type of Berty reactor is shown in Figure 4. The catalyst is contained within a cruciform basket formed from fine wire mesh. Recirculation of the gas is achieved by means of a magnetically driven impeller. Suitable baffling around the periphery of the chamber containing the basket is designed to promote complete mixing.



FIGURE 4 THE BERTY REACTOR

Since the reactor is supposed to behave like a CSTR, the rate of reaction is given directly in terms of the integral conversion measured over the reactor, i.e.



In spite of the widespread publicity of the Berty reactor serious shortcomings in its design were demonstrated by Caldwell [Ref. 5]. These remain to be resolved. The alternative proposed by Caldwell appears to achieve acceptable rates of interphase mass transfer, but temperature gradients can exist within the bed [Ref. 6]. This type of reactor is unsuited to the use of powdered catalyst, so it fails to address the important question of intraparticle gradients (pore diffusion). 5.1.2.4 Catalyst deactivation It would be wrong in passing not to mention the problem of catalyst deactivation. All catalysts deactivate to a greater or lesser extent by one or more of several mechanisms such as thermal sintering, poisoning by trace impurities (e.g. sulfur) or fouling due to "coke" formation. A central feature of the laboratory study may be to isolate the mechanism of deactivation in a particular case and even to quantify its rate. If an integral tubular reactor were used, the poison or coke precursors may often be adsorbed preferentially on the upstream portions of the bed. A band of deactivated catalyst moves along the bed with increasing time in a chromatographic effect while the remainder of the bed remains active. The results may thus be difficult to interpret. CSTR reactors are better suited for studying catalyst deactivation because of their gradientless conditions. Ingenious extensions to their design permit the on-line measurement of coke deposition through thermal gravimetry [Ref. 7].



5.1.2.5 Reactor Ratings To summarize what has been said the different types of reactor are rated good, fair or poor according to the requirements stated at the beginning of 5.1.2 in Table 4 below. TABLE 4 REACTOR RATINGS

An excellent review of multiphase laboratory reactors and their limitations is given by Weekman [Ref. 8].

5.2 Tests for Transport Limitations

In order to ensure that kinetic data obtained in a laboratory reactor reflect truly the chemistry of the reactions concerned, gradients (both thermal and mass) should ideally be eliminated within the particles (intraparticle) and between the external surface of the particles and the adjacent fluid (interphase). Additionally, temperature gradients between the local fluid regions or catalyst particles (interparticle) must be removed, and interparticle mass dispersion may need to be measured separately. Mears [Ref.9] has studied this problem in some depth and draws the following conclusions:

5.2.1 Heat Transfer

As far as temperature gradients are concerned, the order of importance is



Elimination of interparticle temperature gradients guarantees isothermality on a local level As well. Tube diameter is the major variable for consideration in restricting the radial temperature gradient. When the heat transfer resistance at the wall is significant, Mears shows that for a 5% deviation in the rate of reaction across the bed radius,

where λr,eff is the effective thermal conductivity of the bed and Biw, the wall Biot number, is a measure of the ratio of the thermal resistance of the bed to that of the wall. These two parameters are considered in more detail in GBHE-PEG-RXT-810.

Minimizing interparticle temperature gradients is most easily brought about by limiting dt, rather than the reaction rate per unit bed volume, rb, (by catalyst dilution, for example). 5.2.2 Mass Transfer

The picture is reversed for concentration gradients, i.e.

Thus, if pore diffusion can be eliminated by restricting the particle diameter dp, both interphase and interparticle mixing effects are usually negligible, although the axial dispersion problem becomes more pronounced with increasing conversion or reaction order, or with decreasing Peclet numbers (i.e. molecular diffusion). Direct tests for intraparticle and interphase mass transfer resistances are summarized in Table 5.



TABLE 5 EXPERIMENTAL TESTS FOR INTRAPARTICLE AND INTERPHASE GRADIENTS

Note: The range of dp commonly used in a Berty reactor is limited by basket dimensions on the one hand (large dp) and pressure drop across the bed (small dp) on the other. Consequently, the usual test of reducing dp progressively is not very satisfactory. Berty suggests the alternative above. It is also given in Rase's book [Ref. 10].

The intraparticle gradient test for differential and integral plug flow reactors is well established. Smaller particle sizes are usually produced by crushing. Since particle size tends to be fixed in the CSTR arrangements, a rather convoluted test has to be carried out. This involves increasing the total pressure while keeping the reactant partial pressure fixed. If diffusion within the catalyst is bulk controlled, the increase in pressure leads to a decrease in the effective diffusivity, and thus a decrease in observed rate. However, one should be aware that the changes in partial pressure of other components could affect the intrinsic reaction rate. Where diffusion is Knudsen controlled (i.e molecule-pore wall collisions) this test fails completely, even though the reaction may be diffusion limited.

One would also expect to see the apparent activation energy decrease by a factor of two in moving from a chemical to a diffusion rate controlled regime.



It is also expedient to treat the interphase gradient tests with some caution. At low Reynolds number i.e. Re = G dp /µ <1, as might occur in micro-reactors containing crushed catalyst, the mass transfer coefficient kg appears to be independent of FA, and thus no change in the ratio of the mass transfer rate to the chemical rate is brought about. To reiterate, however, satisfaction of the intraparticle gradient test should ensure that interphase mass transfer cannot intrude.

Clearly increasing the stirrer speed in the CSTR should improve interphase mass transfer, but by-passing can mask this effect.

5.3 Building a Reaction Kinetic Equation

The development of a rate equation that is adequate for design purposes involves iteration around the 5 steps shown in Figure 5. Each step should be given careful thought so as to minimize the number of iterations.



FIGURE 5 STEPS IN BUILDING A REACTION RATE EQUATION

5.3.1 Step 1

The first step requires a decision as to which variables are to be studied.



For example, in determining the rate of the single reaction :

A2 + B C + D

which is known to be essentially irreversible over the range of interest, the partial pressure of all components can potentially affect the rate. There would thus be four composition variables and temperature, making 5 variables to consider at the outset. It may be known independently that the product D does not adsorb on the catalyst, in which case it can be eliminated from consideration. In cases where additional components are present, such as a diluent or a selective poison, which can compete for active sites, additional variables are created. In heterogeneous catalysis, in particular, be wary of neglecting the retarding effects of reaction products, i.e. neglecting terms in the denominator of Langmuir-Hinshelwood expressions.

5.3.2 Step 2

It is important at an early stage to set realistic ranges of the variables so as to avoid large extrapolations at the design stage. Close consultation between chemists and engineering is crucial. Inadmissible combinations of variables on the grounds of safety and operability need to be identified.

5.3.3 Step 3

Draw up an experimental plan, the basic requirement of which is to highlight the primary, interactive and non-linear effects of the variables in the smallest number of experiments. Some kind of statistical design will be required. Bearing in mind the need for accurate and direct measurements of the rate of reaction, the choice of reactor would seem to lie between the differential and CSTR types, with the integral plug flow reactor forming the vital testing, Step 5.



For the experimental plan a central-composite design is preferred [Ref. 11]. This comprises three parts :

(i) A conventional 2k-1 fractional factorial design for k variables at 2 levels.

(ii) At least three identical experiments at the centre-point conditions to estimate the percentage of error on the data. These experiments are spread out through the program in order to check for catalyst deactivation.

(iii) Two out-lier experiments for each variable in order to provide an estimate of the non-linearity of the reaction rate to each variable.

Such a design for two factors (P1 and P2) is shown in Figure 6. It effectively saturates the experimental region of interest.



FIGURE 6 A CENTRAL-COMPOSITE DESIGN FOR TWO FACTORS

5.3.4 Step 4

Follow the heuristic ground rules in Table 6 progressively to build a hyperbolic type rate equation in terms of the dimensionless factors of the form :



where subscript cp represents the centre-point value. By so doing one breaks the interdependence of exponential and pre-exponential terms, thereby removing one of the big problems of non-linear estimation. For example, if it were suspected in the reaction:

A2 + B C + D

that the rate-controlling step involved reaction between dissociatively adsorbed A and gas phase B, with D not adsorbed but C adsorbed as a dimer, the hyperbolic rate model would take the form:

Iterate within Step 4 until an adequate-fitting model containing the smallest number of adjustable parameters has been found. A brief discussion of the basis of this type of expression appears in Appendix A.

5.3.5 Step 5

There is no guarantee that the model is adequate for design purposes. Just fitting the data well is not enough. It is highly desirable, therefore, to subject it to rigorous testing. Preferably, the model should be used to predict the performance of a reactor of different geometry and mixing pattern. For example, if a laboratory CSTR were used for developing the rate equation, an integral tubular reactor would be ideal for testing purposes. As an addition, independent methods, e.g. studies of chemisorption, might be utilized to cast light on the true nature of the reaction, thereby enabling some of the assumptions in Step 4 to be questioned. Further improvements in chemical analysis and laboratory reactor design bring nearer the day when 10% error in rate data are realizable, as opposed to 20% in the present-day for complex multi-product reactions involving difficult analysis.



The exponential model is used as a stepping stone to the more rigorous hyperbolic model. A modified exponential rate model of the type :

is developed using linear regression methods. The problem of finding initial parameter estimates is thereby eliminated. Variables that are not significant in the exponential model can be omitted from further consideration - a great help when stepping to a hyperbolic (i.e mechanistic) model. At the centre-point of the design, all dimensionless temperature and concentration terms become unity. Thus, Equation (6) becomes:

Many of the problems cited above would then disappear. An illustration of this heuristic approach is given by Cropley [Ref. 12]. The intention here is to provide a rapid and simple approach to the development of hyperbolic rate models which overcomes most of the problems that historically have hampered their development. This would release the experimenter into areas of more critical testing and modification.



TABLE 6 HEURISTICS FOR BUILDING A HYPERBOLIC-TYPE RATE EQUATION

6 INTRAPARTICLE DIFFUSION

The need to achieve economic rates of reaction of at least 1 mole/sec m3 catalyst volume necessitates surface areas in the range 0.1 – 10 2 m2/cm3. Such areas can only be achieved by ensuring sub-division of the catalyst into the form of a porous particle. An understanding of the pore system, the types of mechanism and rates of molecular diffusion within each type, is essential to successful scale-up of particle size. Pore diffusion generally is an undesirable intrusion, lowering rates of reaction, adversely affecting selectivity and exacerbating catalyst deactivation.



6.1 Types of Pore System

Pore systems can be divided into three types - lattice, precipitate and pelleting pores, the characteristics of which are summarized in Table 7. TABLE 7 TYPES OF PORE SYSTEM

6.1.1 Lattice Pores Lattice pores are contained in naturally occurring or synthetic zeolites employed in catalytic cracking, xylenes isomerization and methanol to gasoline conversion. They form a regular array of cages (15 - 20Å in diameter) connected by passages 3 - 10Å in diameter, and present a high surface area. Molecules move through the zeolite structure by configurational diffusion - a term coined to lump together the importance of not only translational motion, but also vibrational and rotational modes. There is much confusion regarding the magnitude of diffusion coefficients within zeolite crystals, owing to the difficulties of isolation and measurement of the diffusion mechanism. Reported values of D lie in the range 10-9 - 10-14 m2/s. Since crystal sizes are in the 1-3µm range, L2/D may range from 10-2 - 103 seconds.

6.1.2 Precipitate Pores

Precipitate pores form the spaces between particles of a very fine precipitation of catalytic species and support. At low to moderate pressures gases diffuse by Knudsen diffusion (i.e. molecule-pore wall collisions) at rates of the order 1-5 x 10-7 m2/s. For a catalyst pellet of length 5 mm, L2/D will lie in the range 50 - 250 seconds.



6.1.3 Pelleting Pores Pelleting pores are introduced In the final stages of catalyst forming, which may involve tabletting, pressing, or extrusion of a paste followed by calcination. The extrudate may be in the form of a refractory support, such as α-Al2O3, which is then impregnated with a solution of a salt of the catalyst, dried and reduced. Pelleting pores lie in the range 500 – 104Å and above about 5 bar pressure diffusion is in the bulk regime (i.e. controlled by molecule-molecule collisions). Diffusion rates are dependent upon species and composition as well as temperature, pressure and pellet properties, but tend to lie in the range 4-10 x 10-7 m2/s at moderate pressure. Again, for a pellet of 5 mm, L2/D is in the range 25 - 60 seconds. Catalyst pellets may contain two types of pore systems (bi-modal), for example, types c+b or c+a. When both types of pore system offer comparable diffusion resistance, scale-up becomes particularly difficult.

6.2 The Catalyst Effectiveness Factor

The whole endeavor of engineering is to reduce dauntingly complex problems to manageable proportions by judicious choice of assumptions or simply, inspired guesswork. Diffusion and reaction within a porous catalyst is a problem of considerable complication falling into this category. However, in most circumstances, an adequate representation is possible by the use of effective parameters in a 'smoothed out' continuum model, in which diffusion coefficients, concentrations and rates of reaction are continuous functions of position. Occasionally, the continuum model breaks down and a more detailed approach is needed - for example, in zeolite catalysts containing a bi-modal distribution of pore sizes, or for pellets possessing a dense outer skin as a result of the pelleting process.

6.2.1 The Single, Irreversible, First Order Reaction

The simplest cases of the single irreversible first order reaction A B and the consecutive first order reactions A B C provide suitable vehicles for demonstrating the significant questions. A cottage industry has mushroomed around this subject and is admirably dealt with by Aris [Ref. 13]. For the single irreversible first order reaction occurring within the spherical catalyst pellet of radius a, see Figure 7, a steady state material balance over a differential slice of radius r and thickness dr gives :



FIGURE 7 FIRST ORDER ISOTHERMAL IRREVERSIBLE REACTION WITHIN A CATALYST SPHERE

The effectiveness factor ƞ is defined by :



where y = r/a and u = C/CS. Analytical integration of Equations (7), (8) and (9) gives :

in terms of the single dimensionless parameter ɸe, namely the Thiele modulus :

Figure 7 shows that when ɸe is small the concentration of reactant is everywhere close to its surface value. Diffusion is very fast by comparison to chemical reaction and, by definition, ƞ 1. This is usually not of much practical interest because rates of reaction tend to be too low to be economically viable. On the other hand, when ɸe is large the concentration falls exponentially with distance from the outer surface and the reaction is complete within a thin shell near the surface of the pellet. The process is said to be diffusion-controlled. A loose interpretation shows that if ɸe < 1, the process is chemically rate-controlled, whereas if ɸe > 9, it is diffusion limited. Many industrial catalytic processes operate in the diffusion controlled regime as a consequence of the high reaction rates required. If the scale-up problem is simply concerned with reactant conversion, low catalyst effectiveness factors are not too worrying - they can be offset by over sizing the reactor. This is not the case, however, when yield of desired intermediate is the primary concern.



6.2.2 The Consecutive-Parallel First Order Reaction System

A more interesting and practically relevant problem concerns selectivity to desired product B in the consecutive-parallel scheme :

For the case of a catalyst slab of half-thickness a, the concentrations of A and B within the catalyst are given by [Ref. 13].

We can take the analysis a stage further by noting that for an isothermal plug flow reactor, the inter-particle gas concentrations CAS and CBS are related by the differential equation:

Where ū B and ū A are mean concentrations of B and A within the catalyst pellet obtained by integrating Equations (13) and (14).



After some algebra, the result of integrating Equations (13) to (15) is given as a relation between the integral yield to desired product B (Y12) against the fractional conversion of reactant A (XA):

For slab geometry, the catalyst effectiveness factor for the disappearance of A by the parallel reactions is given by [Ref. 13].

Figure 8 displays the integral yield as a function of reactant conversion for a particular case corresponding to a moderate diffusion effect (ƞ = 0.395). Also shown is the case for no diffusion effect (i.e. ɸ1 0). Diffusion serves to lower yield by as much as 10% under moderate conditions.



FIGURE 8 INTEGRAL YIELD vs CONVERSION SHOWING EFFECT OF PELLET DIFFUSION

At a fixed yield (ie raw materials efficiency) of say 0.7, the effect of pellet diffusion would be to reduce the reactant per pass conversion from 0.88 to 0.1. The consequence of this would be to increase the recycle flow rate by a factor 9! Since the cost of separating and recycling unreacted material is proportional to the recycle flow rate, this would likely have serious implications on production costs and capital cost of separation equipment. On the other hand, at a fixed conversion, diffusion lowers product yield by 10%, thereby increasing the usage and thus cost of raw materials, as well as increasing separation costs. In both scenarios, process performance suffers.



When yield issues are of prime importance, as is often the case, a detailed understanding of the impact of pellet diffusion on the total process and its economics is necessary. This information can set targets for process development and better catalyst systems. Pellet diffusion can also affect the rate of catalyst deactivation, the scope of which is beyond this Guide. However, deactivation and diffusion are treated in detail by Hughes [Ref. 14].

6.2.3 Computer Program – GBHE Proprietary Tools for Reactor Modeling

Consult GBHE-PEG-RXT-819 for information on Proprietary Tools for Reactor Modeling to calculate the effectiveness factor for multi-component diffusion, heat transfer and reaction.

6.3 The Measurement of Effective Diffusivity

Table 7 displays ranges of effective diffusivities observed in different types of pore system which are useful for initial calculations of effectiveness factors. For more accurate work the effective diffusivity must be measured experimentally.

Diffusion measurements are usually carried out in the absence of chemical reaction by using chromatographic techniques applied to single catalyst particles or packed beds, or by measuring steady counter-diffusion fluxes across individual pellets. Theoretical treatments and working methods are described elsewhere [Ref. 15].

6.3.1 Simple Uni-modal Pore Size Distribution

If a catalyst has a simple uni-modal pore size distribution, translation of diffusivity measurements made on non-reacting gases can be securely directed to the reactive case, since the effective diffusivity De can be expressed as the product of the structural specific permeability | and a modified diffusion coefficient D* :

The permeability Ψ is determined from measurements of De by one of the methods given above, together with D* determined by the relationship:



where D is the bulk diffusion coefficient and DK is the Knudsen diffusivity, which depends upon the mean pore radius ṝ,

where ṝ is in metres. The mean pore radius can be measured by standard methods. Table 8 presents experimental diffusivities for helium in nitrogen diffusion within the fine pores (ṝ = 3.5 x 10-9 m) of an amorphous silica-alumina catalyst used in xylenes isomerization [Ref.!3].

TABLE 8 COMPARISON OF PORE DIFFUSIVITIES BETWEEN THE DIFFUSION CELL AND PACKED BED METHODS [Ref. 15]

At ambient temperature and pressure D = 7 x 10-5 m2/s. DK = 2.93 x 10-6 m2/s and, from Equation (19), D* = 2.81 x 10-6 m2/s. It follows from Equation (18) and the packed bed result in Table 7, that:



Given Ψ, and armed with the knowledge that Knudsen diffusion is controlling, the effective diffusivity of o-xylene (M = 106) at 723K is predicted from Equations (18-20) to be 1.33 x 10-7 m2/s. This value was combined with the intrinsic reaction rate constant k to determine the effectiveness factor j as a function of pellet sphere diameter from Equation (11). Predicted and measured effectiveness factors are compared in Figure 9 . Theory and practice match quite well given the uncertainties in parameter values. On commercial beads (Dp = 3.5 - 4mm), ŋ ~ 0.5.



FIGURE 9 PREDICTED AND EXPERIMENTAL EFFECTIVENESS FACTORS

6.3.2 Bi-modal Pore Structures

Industrial catalysts sometimes possess a bi-modal distribution of pores - for example, if a fine porous powder of nearly uniform particle size is pelletized and calcined. At moderate pressures, diffusion within the micropores is of the Knudsen type whereas that within the macropores is dominated by bulk diffusion. This presents a number of problems to the reaction engineer.



Firstly, can the simple continuum model containing the single effective diffusivity De, outlined in Clause 6.2, be used to calculate the effectiveness factor? Secondly, if so, how may De be evaluated? Extrapolation of a measurement of De for a non-reacting gas at atmospheric pressure to that for a reacting gas at, say, 50 bars, is fraught with uncertainty - the controlling diffusion regime may, in fact, change!

A rather elegant solution was proposed by Hugo and Koch [Ref. 16]. The ratio of modified diffusion coefficients for the micropores (D*m) and macropores (D*M) may be written as :

From equation (19): At very high pressures, D/DK 0 and Z 1 with:

The structural specific permeability ΨG in Equation (22) contains the contributions of all pores, irrespective of size, since bulk diffusion controls in every pore. On the other hand, at very low pressures D/DK 0 and z dm/d M, the ratio of micropore to macropore diameters. Since this ratio is usually very small, it is permissible to write :

These two limiting forms of equation for De are similar to the trivial uni-modal case. Neither requires assumptions about the arrangement of micropores and macropores.



Hugo and Koch suggest an interpolating form of relation for De to span the whole pressure range in the form :

where Ψ(z) depends upon pressure, and is such that Ψ(o) = ΨM and Ψ(1) = ΨG. These authors found a linear correlation between Ψ and z to be of general validity; i.e. for all porous materials with typical bi-modal pore size distributions. Some of their data are shown in Figure10. They show that Ψ may vary by as much as a factor 3 and always increases with pressure. Their results suggest that two measurements of De are sufficient to locate Ψ at any intermediate pressure - one at normal pressure using a non-reactive gas and, preferably, a second using a liquid to simulate and simplify measurement employing a high pressure gas. Park and Kim's study [Ref. 17] of reaction within biporous catalysts suggests a note of caution in the use of effective diffusivities. They claim that diffusivities measured under reaction conditions can be an order of magnitude lower than those found for non-reactive gases.



FIGURE 10 STRUCTURAL PERMEABILITY vs PRESSURE PARAMETER Z FOR BI-MODAL SUPPORTS

7 ENHANCEMENT OF INTRAPARTICLE DIFFUSIVITY

The flow of material through a catalyst packed bed reactor will require a pressure drop in the direction of flow. If the catalyst particles have macro-pores which penetrate through the particles, this pressure drop will cause forced convection of fluid through the pores which are aligned in the direction of flow.



Nir and Pismen [Ref. 18] have analyzed the case of an irreversible first order liquid phase reaction accompanied by pressure driven flow and diffusion in a porous catalyst slab. Their main findings are shown in Figures 11 and 12. These give the usual effectiveness factor vs Thiele modulus diagram with an additional parameter n where :

an intraparticle Peclet number, where :

Vo = linear velocity in pore L = half the pore length D = diffusivity of limiting reactant within the pore.

The maximum effect of convection (λ) is seen to occur when moderate diffusional effects (1 < Ø < 10) are apparent. The pore velocity Vo is related to the pressure gradient across the pore ∆P/2L by:

In the case of a single tube of circular cross-section :



Where: d t = diameter of the tube In a porous medium we would also expect K to be proportional to the square of the pore diameter. It is best to measure K, but an approximation may be made using pore diameter distribution data by calculating an average value for d from:

Where ni is the number of pores of diameter di.



FIGURE 11 EFFECTIVENESS FACTOR vs THIELE MODULUS AND INTRAPARTICLE PECLET NUMBER



FIGURE 12 RELATIVE INCREASE IN CATALYST PERFORMANCE



8 NOMENCLATURE All effective transport parameters are defined in terms of total areas (void + non-void) normal to the direction of transport



9 BIBLIOGRAPHY

1 Smith, J.M., "Chemical Engineering Kinetics", McGraw-Hill, Tokyo, 2nd Edition (1970).

2 Satterfield C.N. "Mass Transfer in Heterogeneous Catalysis" MIT

Press (1970).

3 Orr, N.H., Cresswell, D.L. and Edwards D.E. "Model Building in Complex Catalytic Reaction Systems. A Case Study: p-Xylene Manufacture", I&EC Proc. Design and Development, (1983) 22, 135.

4 Chinchen C.C., Logan, R.H. and Spencer, M.S. "Water-Gas Shift

Reaction over an Iron oxide/Chromium oxide catalyst. I: Mass Transport Effect". Applied Catalysis (1984) 12,69.

5 Caldwell, L. "An Improved Internal Gas Recirculation Reactor for

Catalytic Studies", Applied Catalysis (1983) 8, 199.

6 Gut, G., Jaeger, R., "Kinetics of the Catalytic Dehydrogenation of Cyclohexanol to Cyclohexanone on a Zinc Oxide catalyst in a Gradientless Reactor", Chem. Eng. Sci. (1982), 37, 319.

7 Sundaram, K.M., Froment, G.F., "Kinetics of Coke Deposition in the

Thermal Cracking of Propane", Chem. Eng. Sci. (1979) 34, 635.

8 Weekman, V.W. Jr., "Laboratory Reactors and their Limitations". A.I.Ch.E.J. (1974) 20, 833.



9 Mears, D.E., "Tests for Transport Limitations in Experimental Catalytic Reactors", I&EC Proc. Design and Development (1971) 10.

10 Rase, H.F. "Chemical Reactor Design for Process Plants" John

Wiley (1977).

11 Davies, O.L., "Design and Analysis of Industrial Experiments", 534-5, Hafner Publishing Co. New York. (1960).

12 Cropley, J.B. "Heuristic Approach to Complex Kinetics", A.C.S.

Symposium Series No.65, Chemical Reaction Engineering - Houston p292 (1978).

13 Aris, R. "The Mathematical Theory of Diffusion and Reaction in

Permeable Catalysts", vol 1: The Theory of the Steady State", Oxford University Press (1975).

14 Hughes, R. "Deactivation of Catalysts", Academic Press (1984). 15 Cresswell, D.L. and Orr, N.H. "Measurement of Binary Gaseous

Diffusion Coefficients within Porous Catalysts" from Residence Time Distribution Theory in Chemical Engineering, edited by A. Petho and R.D. Noble, Verlag Chemie, Weinheim, p.41 (1982). IC 07039/C Cresswell, D.L. Simultaneous sorption and diffusion within adsorbent granules using pulse chromatography. (Nov 1986).

16 Hugo, P. and Koch, E.N. "Diffusion within Bi Porous Catalysts",

German. Chem. Eng. (1985) 8, 234. 17 Park, S.H. and Kim, Y.G. "The Effect of Chemical Reaction on

Effective Diffusivity within Bi!Porous Catalysts - II Experimental Study". Chem. Eng. Sci. (1984) 39, 533.

18 A. Nir and L.M. Pismen "Simultaneous intraparticle forced

convection, diffusion and reaction in a porous catalyst." Chem. Eng. Sci. (1977) 32, 35.

9.1 Further Reading

Berty, J.M., "Reactor for Vapour phase Catalytic Studies", Chem. Eng. Prog. (1974) 70,578.



Jackson R, "Transport Processes within Porous Catalysts", Elsevier Amsterdam (1977).

Satterfield C.N. "Heterogeneous Catalysis in Practice" McGraw-Hill (1980).

Sharma, R.K., Cresswell, D.L. and Newson, E.J. Selective Oxidation of Benzene to Maleic Anhydride at Commercially Relevant Conditions", I.S.C.R.E. 8 p353, Edinburgh, September (1984); Inst. Chem. Eng. Symp. Series No.87.



APPENDIX A LANGMUIR - HINSHELWOOD KINETICS

The Langmuir-Hinshelwood form of rate expression; e.g. see 5.1.1, arises out of the mechanism of solid catalyzed reactions. This mechanism is simplified as :

(a) adsorption of reactants onto the catalyst surface;

(b) reaction of (adsorbed) species;

(c) desorption of products.

A simple example will illustrate. Consider the solid catalysed reaction between A and B to produce C and D.

A + B C + D

Steps (a) would be written :

A + s As B + s Bs

Here s is a vacant catalyst active site As and Bs are adsorbed species

Step (b) reaction between two adsorbed species :

As + Bs Cs + Ds

Steps (c)

Cs C + s Ds D + s

If we now consider concentrations X, we could write physical equilibrium equations:



If we let XS be the total "concentration" of available catalyst active sites, at physical equilibrium we can write:

Now if step (b) were rate controlling and the adsorption steps were at equilibrium, the rate expression might be :



If adsorption of A were rate controlling, the rate expression might be :

Notice that the KA X A term has disappeared from the denominator because the reaction consumes A as fast as the adsorption step can supply it, i.e. XAs is near zero, it is not in equilibrium. If a desorption step were rate controlling, the catalyst would be flooded with the limiting product and the reaction rate could be apparently zero order.



A variety of different rate expressions results from different concepts of the reaction mechanism.

DOCUMENTS REFERRED TO IN THIS PROCESS ENGINEERING GUIDE

This Process Engineering Guide makes reference to the following documents: GBHE ENGINEERING GUIDES Glossary of Engineering Terms (Referred to in Clause 3).

GBHE-PEG-RXT-800 Guide on How to use the Reactor Technology

Guides (Referred to in 4.1) GBHE-PEG-RXT-802 Residence Time Distribution Data

(Referred to in Clause 4 and 4.1) GBHE-PEG-RXT-809 Homogeneous Reaction - Gas Solid Systems

(Referred to in 5.2.1) GBHE-PEG-RXT-812 Case Studies in Reactor Technology

(Referred to in Clause 5)

GBHE-PEG-RXT-817 Tools for Reactor Modeling

Part C (referred to in 4.1 and 6.2.3)

Part F (referred to in 5.1.1).