Catalyst particle model

For the catalyst particle modeled by the network this ratio corresponds to the ratio of the geometric area to overall surface area, which for most catalysts is essentially zero as is also the case for three-dimensional networks. Accounting for the activity of the surface pores in Bethe networks tends to smooth out all the abrupt changes in the activity that would be otherwise observed at the percolation threshold. 

Weidman, D. L. Catalyst particle modeling in fixed-bed reactors, Ph. D. thesis, University of Wisconsin-Madison (1990). 

Simulation of tubular steam reformers and a comparison with industrial data are shown in many references, such as [250], In most cases the simulations are based on measured outer tube-wall temperatures. In [181] a basic furnace model is used, whereas in [525] a radiation model similar to the one in Section 3.3.6 is used. In both cases catalyst effectiveness factor profiles are shown. Similar simulations using the combined two-dimensional fixed-bed reactor, and the furnace and catalyst particle models described in the previous chapters are shown below using the operating conditions and geometry for the simple steam reforming furnace in the hydrogen plant. Examples 1.3, 2.1 and 3.2. Similar to [181] and [525], the intrinsic kinetic expressions used are the Xu and Froment expressions [525] from Section 3.5.2, but with the parameters from [541]. 

However, the molecular reaction did not consider the temperature variation, as well as the internal and external diffusion during the reaction. Therefore, due to the integral accumulation effect along the catalyst bed, the modified ASF distribution has appeared, even though the reactor is strict isothermal and the internal and external diffusion of catalyst can be ignored. Therefore, if we want to control the product distribution of FTS, we must clarify the detailed dynamic information on the particle-reactor level, which needs to couple the catalyst particle model and the reactor flow model. 

X being the average displacement of the particles in the time t. The obvious difference between these colloidal dispersions aird the catalyst particles on the surface of a support, is that the above model would require that the particles... 

Since regular helices with the inner layer matching the catalyst particle size have been observed[4,5], we propose a steric hindrance model to explain the possible formation of regular and tightly wound helices. 

Model based on the variation of the active catalyst perimeter. To form the (5,5)-(9,0) knee represented in Fig. 13(c) on a single catalyst particle, the catalyst should start producing the (5,5) nanotubule of Fig. 13(a), form the knee, and afterwards the (9,0) nanotubule of Fig. 13(b), or vice versa. It is possible to establish relationships between... 

Fig. 13. Model of the growth of a nanotubule bonded to the catalyst surface, (a) Growth of a straight (5,5) nanotubule on a catalyst particle, with perimeter I5ak (b) growth of a straight (9,0) nanotubule on a catalyst particle whose perimeter is 18ak (k is a constant and the grey ellipsoids of (a) and (b) represent catalyst particles, the perimeters of which are equal to 5ak and 18a/t, respectively) (c) (5,5)-(9,0) knee, the two sides should grow optimally on catalyst particles having perimeters differing by ca. 20%.

Model based on the variation of the number of active" coordination sites at the catalyst surface. The growth of tubules during the decomposition of acetylene can be explained in three steps, which are the decomposition of acetylene, the initiation reaction and the propagation reaction. This is illustrated in Fig. 14 by the model of a (5,5) tubule growing on a catalyst particle ... 

As our first approach to the model, we considered the controlling step to be the mass transfer from gas to liquid, the mass transfer from liquid to catalyst, or the catalytic surface reaction step. The other steps were eliminated since convective transport with small catalyst particles and high local mixing should offer virtually no resistance to the overall reaction scheme. Mathematical models were constructed for each of these three steps. 

Farkas and Sherwood (FI, S5) have interpreted several sets of experimental data using a theoretical model in which account is taken of mass transfer across the gas-liquid interface, of mass transfer from the liquid to the catalyst particles, and of the catalytic reaction. The rates of these elementary process steps must be identical in the stationary state, and may, for the catalytic hydrogenation of a-methylstyrene, be expressed by ... 

Semibatch Model "GASPP". The kinetics for a semibatch reactor are the simpler to model, in spite of the experimental challenges of operating a semibatch gas phase polymerization. Monomer is added continuously as needed to maintain a constant operating pressure, but nothing is removed from the reactor. All catalyst particles have the same age. Equations 3-11 are solved algebraically to supply the variables in equation 5, at the desired operating conditions. The polymerization flux, N, is summed over three-minute intervals from the startup to the desired residence time, t, in hours ... 

Step 3. Transport within a catalyst pore is usually modeled as a one-dimensional diffusion process. The pore is assumed to be straight and to have length The concentration inside the pore is ai =ai(l,r,z) where I is the position inside the pore measured from the external surface of the catalyst particle. See Figure 10.2. There is no convection inside the pore, and the diameter of the pore is assumed to be so small that there are no concentration gradients in the radial direction. The governing equation is an ODE. 

Traditionally, an average Sherwood number has been determined for different catalytic fixed-bed reactors assuming constant concentration or constant flux on the catalyst surface. In reality, the boundary condition on the surface has neither a constant concentration nor a constant flux. In addition, the Sh-number will vary locally around the catalyst particles and in time since mass transfer depends on both flow and concentration boundary layers. When external mass transfer becomes important at a high reaction rate, the concentration on the particle surface varies and affects both the reaction rate and selectivity, and consequently, the traditional models fail to predict this outcome. 

Above we considered a porous catalyst particle, but we could similarly consider a single pore as shown in Fig. 5.36. This leads to rather similar results. The transport of reactant and product is now determined by diffusion in and out of the pores, since there is no net flow in this region. We consider the situation in which a reaction takes place on a particle inside a pore. The latter is modeled by a cylinder with diameter R and length L (Fig. 5.36). The gas concentration of the reactant is Cq at the entrance of the pore and the rate is given by... 

Fixed-bed reactors are used for testing commercial catalysts of larger particle sizes and to collect data for scale-up (validation of mathematical models, studying the influence of transport processes on overall reactor performance, etc.). Catalyst particles with a size ranging from 1 to 10 mm are tested using reactors of 20 to 100 mm ID. The reactor diameter can be decreased if the catalyst is diluted by fine inert particles the ratio of the reactor diameter to the size of catalyst particles then can be decreased to 3 1 (instead of the 10 to 20 recommended for fixed-bed catalytic reactors). This leads to a lower consumption of reactants. Very important for proper operation of fixed-bed reactors, both in cocurrent and countercurrent mode, is a uniform distribution of both phases over the entire cross-section of the reactor. If this is not the case, reactor performance will be significantly falsified by flow maldistribution. 

The reactor system works nicely and two model systems were studied in detail catalytic hydrogenation of citral to citronellal and citronellol on Ni (application in perfumery industty) and ring opening of decalin on supported Ir and Pt catalysts (application in oil refining to get better diesel oil). Both systems represent very complex parallel-consecutive reaction schemes. Various temperatures, catalyst particle sizes and flow rates were thoroughly screened. 

A useful application of the model is to examine the S02 and 02 concentration profiles in the trickle bed. These are shown for the steady-state conditions used by Haure et al. (1989) in Fig. 25. The equilibrium S02 concentration drops through the bed, but the 02 concentration is constant. In Haure s experiments 02 partial pressure is 16 times the S02 partial pressure. At the catalyst particle surface, however, 02 concentration is much smaller and is only about one-third of the S02 concentration. This explains why 02 transport is rate limiting and why experimentally oxidation appears to be zero-order in S02. 

Scanning electron microscopy and other experimental methods indicate that the void spaces in a typical catalyst particle are not uniform in size, shape, or length. Moreover, they are often highly interconnected. Because of the complexities of most common pore structures, detailed mathematical descriptions of the void structure are not available. Moreover, because of other uncertainties involved in the design of catalytic reactors, the use of elaborate quantitative models of catalyst pore structures is not warranted. What is required, however, is a model that allows one to take into account the rates of diffusion of reactant and product species through the void spaces. Many of the models in common use simulate the void regions as cylindrical pores for such models a knowledge of the distribution of pore radii and the volumes associated therewith is required. 

Because of the inadequacies of the aforementioned models, a number of papers in the 1950s and 1960s developed alternative mathematical descriptions of fluidized beds that explicitly divided the reactor contents into two phases, a bubble phase and an emulsion or dense phase. The bubble or lean phase is presumed to be essentially free of solids so that little, if any, reaction occurs in this portion of the bed. Reaction takes place within the dense phase, where virtually all of the solid catalyst particles are found. This phase may also be referred to as a particulate phase, an interstitial phase, or an emulsion phase by various authors. Figure 12.19 is a schematic representation of two phase models of fluidized beds. Some models also define a cloud phase as the region of space surrounding the bubble that acts as a source and a sink for gas exchange with the bubble. 

Modeling of Jet-Induced Attrition. Werther and Xi (1993) compared the jet attrition of catalysts particles under steady state conditions with a comminution process. They suggested a model which considers the efficiency of such a process by relating the surface energy created by comminution to the kinetic energy that has been spent to produce this surface area. The attrition rate, RaJ, defined as the mass of attrited and elutriated fines per unit time produced by a single jet, is described by... 

The focus of the remainder of this chapter is on interstitial flow simulation by finite volume or finite element methods. These allow simulations at higher flow rates through turbulence models, and the inclusion of chemical reactions and heat transfer. In particular, the conjugate heat transfer problem of conduction inside the catalyst particles can be addressed with this method. 

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