Catalytic effectiveness of porous particles: A continuum analytic model including internal and external surfaces
Graphical abstract
Introduction
The pioneering concept of catalytic activity on porous structures was independently published in the late 1930s by Thiele in the United States (Thiele, 1939), Damkohler in Germany (Damkohler, 1937), and Zeldovitch in Russia (Zeldovitch, 1939). They formulated a pseudo-continuum model of coupled mass transfer and reaction kinetics in a porous structure, and introduced the well-known effectiveness factor estimator for the porous catalyst. One of their important assumptions was the negligibility of the external surface. According to Thiele, “the greater part of the surface available for reaction is assumed to be on the walls of the pores in the catalyst. The actual external surface is assumed to be negligible in comparison” (Thiele, 1939). Under this assumption, the reactions and heat production/consumption on the external surface of the porous particle are both ignored.
This assumption is valid for many applications. For example, a typical catalyst particle with a modest specific surface area of 50 m2/g (Salmi et al., 2011), a diameter of 0.5 cm, and a weight of 0.1 g has an external to total surface area ratio as small as ≈10−5. A thin outer catalyst shell of ≈10−5 cm in thickness would contain the same surface area as the external surface. This shell will emulate the external 2D surface in the pseudo-continuum volume model. Hence, this assumption is commonly realized and is taken for granted in most published research articles (Bischoff, 1965, Gottifredi and Gonzo, 2005, Kim and Lee, 2006) and textbooks (Davis and Davis, 2013, Froment et al., 2011, Levenspiel, 1999, Mann, 2009, Salmi et al., 2011).
In some systems, however, the external area is not negligible as was pointed out by Farcasiu and Degnan (1988). Examples include silver-catalyzed partial oxidation of ethylene (Varghese et al., 1978), Zr2Fe non-evaporable hydrogen getter (Cohen et al., 2016), and materials with functionalized internal and external surfaces (French et al., 2004). Moreover, porous nano-sized particles have been made possible due to the advances in industrial catalyst synthesis (Valtchev and Tosheva, 2013). Their large surface-to-volume ratio means that the external surface is also expected to be important.
Hence, the external surface clearly can and should be included in an effectiveness model for certain porous materials. In the words of Thiele, “it is obvious that there may be all degrees between smooth platinum or nickel and a very porous material” (Thiele, 1939). Such a model should remain valid for the two limiting cases of bulk and highly porous materials (with the external to total surface area ratio being one and zero, respectively), and intermediary cases. Nevertheless, only a few attempts have been made to formulate a descriptive analytic model that explicitly accounts for the external surface, and they have achieved only partial success. Catalysis on the external and internal surfaces were treated separately, and the results were then combined algebraically to obtain the composite expression for the effectiveness and yield (Farcasiu and Degnan, 1988, Goldstein and Carberry, 1973, Kramer, 1966). However, only a simultaneous account of reactions on both surfaces can reproduce the correct behavior and asymptotic limits.
Varghese et al. (1978) formulated an external-internal surface coupled model. Although the presented analytic solution limits to the case with negligible external surface, it does not satisfy the model boundary conditions. Moreover, the model itself doesn't limit to the case of negligible external surface since the external surface source-term in the boundary condition does not vanish for negligible external surface (Varghese et al., 1978). Fraenkel (1990) developed a different, simple kinetic model that explains isomer shape selectivity in zeolite catalysts. However, the model was based on simplifying the system using specific kinetic and asymptotic assumptions.
Most of the textbooks on reaction chemistry and chemical engineering bypass the subject of external surface when describing the effectiveness of porous materials, and take Thiele's assumption for granted (Davis and Davis, 2013, Froment et al., 2011, Levenspiel, 1999, Mann, 2009, Salmi et al., 2011). While discussing the overall effectiveness factor, Fogler starts with a detailed formulation for the external and internal surfaces but ends with the application of Thiele's assumption. Moreover, the total catalytic surface area is erroneously used when formulating the area of the internal surface (Fogler, 2016).
Additionally, the topic of non-uniform catalyst distribution has been studied extensively in the quest to optimize catalyst activity. Pellets with different material profiles and consisting of various solid species were examined. However, none of the suggested models explicitly account for the external surface (Au et al., 1995, Melchiori et al., 2015, Morbidelli et al., 1982).
In this study, we present a general, steady state, descriptive model for the reaction of A → B in the case of a porous particle with internal and external mass transfer limitation, explicitly accounting for the external to total surface area ratio (Fig. 1). We then solve the model for an isothermal first-order reaction in a porous sphere, and compare the analytic results to the well-known effectiveness factor where external surface was neglected (Thiele, 1939).
Although the computational resources nowadays permit very detailed numerical modeling (Andersen and Evje, 2016, Pereira et al., 2014), we believe that a simple analytic model, such as the one presented here, enables a better understanding of the role of external surface in porous materials that is relevant in some cases.
Section snippets
Actual differences between external and internal surface areas
First, we define the concept of surface area and related terms. In the ideal theoretical case, the particle surface is perfectly smooth (no surface roughness), and the external surface area equals the geometric area. The ratio between the volume and the geometric external surface area is defined as the characteristic length, a. The geometric volume and surface can be calculated from the weight and density of the pellets (V = mp/ρp, S = f(V)).
The specific surface area is the total (internal and
Discussion
Based on the concentration profile (Eq. (9)) and the relevant reaction terms (Eqs. (4), (5)), we have calculated the system's global catalytic effectiveness factor ():
Since the effectiveness is analogous to normalized conductance, the reciprocal catalytic effectiveness can be considered a kind of resistance. Hence, after Aris (Hatfield and Aris, 1969), can be alternatively presented as a network of
Conclusions
In the present study, we extended the well-known pseudo-continuum model of porous catalyst particles to explicitly include the effect of the external surface. In accordance with Thiele's expectation, the developed model is continuous and correctly asymptotes to the well-known limiting cases of bulk particle (only external surface) and highly porous particle (negligible external surface). Previous attempts to estimate the global effectiveness factor using the sum of the internal and external
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