The New Narrative Approach

Coke is hydrogen-deficient carbonaceous residues deposited on the surface. It is considered to be formed by a condensation polymerization which eventually leads to the formation of such a large polymer structure as to block the active sites on the catalyst surfaces.³⁹ For instance, in catalytic cracking the analysis of a coke deposit on a used cracking catalyst indicated a mixture of solid and semiliquid mixture of polynuclear aromatics, such as dimmers and trimers of naphthalene, phenanthrene, etc.⁴⁰ Besides the form of hydrogen-deficient polymers or aromatics, in some reactions the element carbon can form coke, which includes the metal carbide phase of Fisher-Tropsch synthesis on iron-based catalysts and the filamentous phase for steam reforming of methane on nickel-based catalysts.³⁹

Coke formation is a complicate process that oversimplified empirical correlation obtained by Voorhies⁴¹ from the cracking of gas oil feedstock has been widely accepted.

with 0.5 < n < 1

n

CC =At (2.6)

where t is the process time and A and n are constants. The values of n were determined for different reactions. Voorhies postulated that the rate of coke formation was controlled by diffusion mechanism and not dependent on the space time; the diffusion rate could be expressed as inversely proportional to the weight percent of carbon deposited. Ozawa and Bischoff⁴² used the thermogravimetric method to measure the weight of coke formed on catalyst for the cracking of ethylene over a silica-alumina catalyst for various process times. They found that a simple empirical correlation was

process time. Also Eberly et al.⁴³ showed that the production of coke in fixed beds over wide space velocities was not completely independent of space velocity. In general, the correlation, Eq. (2.6), has been used in many systems over the years for its simplicity.

However, the origin of coke was totally neglected.

A theoretical and mechanistic approach of kinetic modeling of coke formation was first investigated by Froment and Bischoff.^{44, 45} Froment and Bischoff⁴⁴ pointed out that the rate of coke formation can not be established without taking into account the rate of main reaction, since coke is formed, definitely, from the reaction mixture. Two activity functions, i.e., an exponential dependence of the catalyst activity on the coke content and a hyperbolic dependence on the coke content, were introduced to show the effect of the coke on the catalyst activity.

Deactivation functions are defined as the ratio of rates of a chemical reaction for the main reaction:

Aio Ai Ai

r

r = Φ (2.7)

where r is the initial reaction rate in absence of coke. _Ai^o Deactivation function for the coke formation is

C o C C

r

r = Φ (2.8)

where r is the initial coking rate. Therefore, the rate equation of coke formation is _C^o given by

C o

C C

dC r

dt = Φ (2.9)

The initial coking rate, r , is a function of operating conditions, i.e., temperature and _C^o partial pressures. The following deactivation functions were suggested by Dumez and Froment.⁴⁶

Numerous investigations for the kinetic modeling of coke have been conducted by Froment and co-workers. Examples are: isomerization of pentane on the reforming catalyst,⁴⁷ steam/CO2 reforming of methane,^{48, 49} steam cracking,⁵⁰ dehydrogenation of 1-butene into butadiene,^{46, 51} and dehydrogenation of ethylbenzene into styrene.⁵² Reviews for a rigorous formulation of a kinetic model of coke formation were presented by Froment.^{53, 54}

For the main reaction A→B, the rate is written involved. Generally, if the main reaction involves nA sites in the rate determining step, then the deactivation function ϕ is formulated as _A

nA

Since a coking reaction itself is also deactivated by the coke, the rate of coke formation can be described by

0 1

In the same way as Eq. (2.13) the deactivation function is given by

nC

The approach explained here relates the deactivation functionϕ to the coke content CC, namely φ = f(Cc). De Pauw and Froment⁵⁵ and Dumez and Froment⁴⁶ derived an

exponential relationship between deactivation function and coke content, which was determined by means of an electrobalance. An electrobalance is the primary equipment for the kinetic analysis of coke formation. The literature regarding this can be found in Ozawa and Bischoff for ethylene cracking,⁴² Wagner and Froment for methane steam reforming,⁵⁶ Beirnaert et al. for catalytic cracking of n-hexane,⁵⁷ and Snoeck et al. for methane cracking.⁵⁸

2.5.3 Deactivation by Site Coverage and Pore Blockage

If coke growth and pore blockage are involved in the coking mechanism, Eqs.

(2.13) and (2.16) are no longer valid with respect to the definition of the deactivation functions in Eqs. (2.11) and (2.14), respectively. Beeckman and Froment^{59, 60} investigated this situation. They treated the deactivation by site coverage and pore blockage using probability functions. The internal structure of the particle was first assumed to be a single pore. The deactivation function depended on the textural properties of catalyst and physical properties of coke. Marin et al.⁵¹ explained the deactivation by coke deposition in butene dehydrogenation on Cr2O3/Al2O3 in terms of site coverage and pore blockage.

Beeckman and Froment⁶¹ extended the deactivation study to a stochastic pore network model and considered diffusion, reaction, and deactivation by site coverage only. The pore network was represented by a Bethe-tree in which the pores of catalyst are represented by the bonds of a tree and their intersections are represented by the nodes.

Since the percolation theory, which is a more reliable model to describe the pore

number of studies were made in this area.^63-66 The percolation theory was intensively reviewed by Sahimi et al.⁶⁷