Kinetics tortuosity factor

Many theoretical embellishments have been made to the basic model of pore diffusion as presented here. Effectiveness factors have been derived for reaction orders other than first and for Hougen and Watson kinetics. These require a numerical solution of Equation (10.3). Shape and tortuosity factors have been introduced to treat pores that have geometries other than the idealized cylinders considered here. The Knudsen diffusivity or a combination of Knudsen and bulk diffusivities has been used for very small pores. While these studies have theoretical importance and may help explain some observations, they are not yet developed well enough for predictive use. Our knowledge of the internal structure of a porous catalyst is still rather rudimentary and imposes a basic limitation on theoretical predictions. We will give a brief account of Knudsen diffusion. 

If the effectiveness factor of the catalyst is known to be 0.42, estimate the tortuosity factor of the catalyst assuming that the reaction obeys first-order kinetics and that Knudsen diffusion is the dominant mode of molecular transport. 

From the foregoing discussion, it is clear that the catalyst pellet is not only the heart of the catalytic reactor but also the hardest part of the system to model accurately. The difficulties associated with the modelling of the single pellet (especially the porous pellet) is due to the uncertainties associated with the intrinsic kinetics and the precise modelling of diffusion of mass and heat inside the pellet as well as the complex interaction between these two processes. The complex tortuous structure of porous catalyst pellet adds to the complexity. Different trials to estimate the tortuosity factor (which accounts for the complex tortuous structure of the pellet) theoretically have failed to give accurate results and this factor is usually estimated experimentally. 

Equation (7,17) is called the Kozeny-Carman equation and is applicable for flow through beds at particle Reynolds numbers up to about 1.0. There is no sharp transition to turbulent flow at this Reynolds number, but the frequent changes in shape and direction of the channels in the bed lead to significant kinetic energy losses at higher Reynolds numbers. The constant 150 corresponds to = 2.1, which is a reasonable value for the tortuosity factor. For a given system, Eq. (7,17) indicates that the flow is proportional to the pressure drop and inversely proportional to the fluid viscosity. This statement is also known as Darcy s law, which is often used to describe flow of liquids through porous media. 

Consider the synthesis of methanol from carbon monoxide and hydrogen within the internal pores of catalysts with cylindrical symmetry. The radius of each catalytic pellet is 1 mm, the average intrapellet pore radius is 40 A, the intrapellet porosity is 0.50, the intrapellet tortuosity factor is 2, and the gas-phase molar density of carbon monoxide in the vicinity of the external surface of the catalytic pellet is 3 x 10 g-mol/cm. A reasonable Hougen-Watson kinetic rate law is based on the fact that the slowest step in the mechanism is irreversible chemical reaction that requires five active sites on the catalytic surface, due to the postulate that both hydrogen molecules must dissociate and adsorb spontaneously (see Section 22-3.1). Do not linearize the rate law. In units of g-mol/cm -min-atm, the forward kinetic rate constant is... 

The balances of mass of the chemical species i and the terms for the adsorption kinetics (mass transfer, pore diffusion) are listed in Table 9.5-1 for the three systems with Cj as the concentration in the fluid phase and Xj as the mass loading of the adsorbent. J3 denotes the mass transfer coefficient of a pellet and sj, is its internal porosity. The tortuosity factor will be explained later. The derivation of equations describing instationary diffusion in spheres has already been presented in Sect. 4.3.3. With respect to diffusion in macropores it is important to consider that diffusion can take place in the fluid as well as in the adsorbate phase. In Table 9.5-1 special initial and boimdaty conditions valid for a completely unloaded bed (adsorption) or totally loaded bed (desorption) are given. In this section only the model valid for a thin layer in a fixed bed with the thickness dz and the volmne / dz will be derived, see Fig. 9.5-2. 

The supposition of negligible internal mass transfer resistances was validated by solving Model A using the obtained kinetics parameters, and evaluating the effectiveness factor (Eq.3). The D jf values were estimated assuming tortuosity factors reported in the literature (Hayes and Kolaczkowski, 1997), i.e. the effective diffusivity was not fitted. The solution of the heterogeneous Model A indicates that the reactor operates with effectiveness factors between 1 and 0.98 for the temperature range 310 - 420 °C. 

Xu and Froment (1989b) developed the most widely accepted kinetic model, deriving the intrinsic parameters and incorporating diffusional limitations through the evaluation of the tortuosity factor, effective diffusivities, and the effectiveness factor. These parameters were used in the simulation of commercial reactors and industrial steam reformers with satisfactory results. 

For process modeling proposes the effective chemical reaction rate has to be expressed as a function of the liquid bulk composition x, the local temperature T, and the catalyst properties such as its number of active sites per catalyst volume c, its porosity e, and its tortuosity t. As discussed in Section 5.4.2, the chemical reaction in the catalyst particles can be influenced by internal and external mass transport processes. To separate the influence of these transport resistances from the intrinsic reaction kinetics, a catalyst effectiveness factor p is introduced by... 

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