Cylindrical catalyst pellets diffusion/reaction

Next, the classical problem of diffusion with reaction in a cylindrical catalyst pellet is considered [8] [4]... 

Consider diffusion with a second-order reaction in a cylindrical catalyst pellet (exercise problem 2 chapter 3). Solve this problem using recursion technique described in section 10.1.2. 

Example 12-2 Using the intrinsic rate equation obtained in Example 12-1, calculate the global rate of the reaction o-Hj p- % at 400 psig and — 196°C, at a location where the mole fraction of ortho hydrogen in the bulk-gas stream is 0.65. The reactor is the same as described in Example 12-1 that is, it is a fixed-bed type with tube of 0.50 in. ID and with x -in. cylindrical catalyst pellets of Ni on AljOj. The superficial mass velocity of gas in the reactor is 15 lb/(hr)(ft ). The effective diffusivity can be estimated from the random-pore model if we assume that diffusion is predominately in the macropores where Knudsen diffusion is insignificant. The macroporosity of the pellets is 0.36. Other properties and conditions are those given in Example 12-1. 

Thermal effects constitute a significant portion of the study devoted to catalysis. This is true of electrochemical reactions as well. In general the reaction rate constants, diffusion coefficients, and conductivities all exhibit Arrhenius-type dependence on temperature, and as a rule of the thumb, for every 10°C rise in temperature, most reaction rates are doubled. Hence, temperature effects must be incorporated into the parameter values. Fourier s law governs the distribution of temperature. For the example with the cylindrical catalyst pellet described in the previous section, the equation corresponding to the energy balance can be written in the dimensionless form as follows ... 

Figure 6.17.11 shows experimental results of the epimerization at three different temperatures for a small particle size (0.5-1 mm), which represent the intrinsic kinetics, and for the original 6 x 6 mm cylindrical catalyst pellets, where pore diffusion limitations lead to a decrease of the effective reaction rate. The experiments were conducted in the well-mixed batch reactor. As expected, the influence of mass transfer increases with increasing temperature and becomes strong at 200°C. Note that for clarity Figure 6.17.11 only shows the change of menthol concentration and not of the other two stereoisomers (as in Figure 6.17.5). In addition, note that the initial menthol concentration is not zero as an industrially relevant feed was used. The dashed and solid lines in Figure 6.17.11 represent the results of the calculation by the method described before, showing a good agreement with the experimental data.

Wheeler s treatment of the intraparticle diffusion problem invokes reaction in single pores and may be applied to relatively simple porous structures (such as a straight non-intersecting cylindrical pore model) with moderate success. An alternative approach is to assume that the porous structure is characterised by means of the effective diffusivity. (referred to in Sect. 2.1) which can be measured for a given gaseous component. In order to develop the principles relating to the effects of diffusion on reaction selectivity, selectivity in isothermal catalyst pellets will be discussed. 

The catalyst packing of the reactor consists of an iron oxide Fe20s, promoted with potassium carbonate K COo, and chromium oxide Cr O-s,. The catalyst pellets are extrudates of a cylindrical shape. Since at steady state the problem of simultaneous diffusion and reaction are independent of the particle shape, an equivalent slab geometry is used for the catalyst pellet, with a characteristic length making the surface to volume ratio of the slab equal to that of the original shape of the pellet. 

Example 10-1 Experimental, global rates are given in Table 10-2 for two levels of conversion of SOj to SO3. Evaluate the concentration difference for SO2 between bulk gas and pellet surface and comment on the significance of external diffusion. Neglect possible temperature differences. The reactor consists of a fixed bed of x -in. cylindrical pellets through which the gases passed at a superficial mass velocity of 147 lb/(hr)(ft ) and at a pressure of 790 mm Hg. The temperature of the catalyst pellets was 480°C, and the bulk mixture contained 6.42 mole % SOj and 93.58 mole % air. To simplify the calculations compute physical properties on the basis of the reaction mixture being air. The external area of the catalyst pellets is 5.12 ft /lb material. The platinum covers only the external surface and a very small section of the pores of the alumina carrier, so that internal diffusion need not be considered. 

Metal deposition occurs with sharp gradients within a catalyst pellet, usually concentrated on the outside of catalyst pellets forming a U-shaped distribution. Sato et at [3] related this metal deposition with simultaneous diffusion and reaction, and suggested a value of 8 for the Thiele modulus in a slab geometry. Tamm [4] suggested that this distribution can be characterized by a theta factor defined in a cylindrical geometry as... 

The equations for simultaneous pore diffusion and reaction were solved independently by Thiele and by Zeldovitch [16,17]. They assumed a straight cylindrical pore with a first-order reaction on the surface, and they showed how pore length, diffusivity, and rate constant influenced the overall reaction rate. Their solution cannot be directly adapted to a catalyst pellet, since the number of pores decreases going toward the center and assuming an average pore length would introduce some error. The approach used here is that of Wheeler [18] and Weisz [19], who considered reactions in a porous sphere and related the diffusion flux to the effective diffusivity, Z). The basic equation is a material balance on a thin shell within the sphere. The difference between the steady-state flux of reactant into and out of the shell is the amount consumed by reaction. 

Porous, cylindrical-shaped pellets are used as catalyst for the reaction A products, in a packed bed. We wish to model the steady-state diffusion-reaction processes within the particle. When the pellet length to diameter ratio (L/2R) > 3, flux to the particle end caps can be ignored. Assume the surface composition is and that reaction kinetics is controlled by a linear rate expression = kC (mole/volume time), and diffusive flux obeys = -D dC /dr. 

Diffusion and Reaction in a Single Cylindrical Pore within the Catalyst Pellet... 

If a catalyst pellet (of any shape) has well-structured pores that are of imiform diameter d and length L and the pores are uniformly distributed throughout the volume of the pellet, then the overall rate equation can be derived by accounting for the rate of diffusion and rate of reaction in one single pore within the catalyst pellet. Consider a cylindrical pore of diameter d and length L (Figure 4.24) in a catalyst pellet in contact with a gas stream containing reactant A at concentration Ag- AS is the concentration of A in the gas at the pore mouth on the outer surface of the catalyst pellet. 

First, the potential exhibits a maximum or a minimum at a point or axis of symmetry. These locations can be the centerline of a slab, the axis of a cylinder, or the center of a sphere. Figure 1.2a and Figure 1.2b consider two such cases. Figure 1.2a represents a spherical catalyst pellet in which a reactant of external concentration Q diffuses into the sphere and undergoes a reaction. Its concentration diminishes and attains a minimum at the center. Figure 1.2b considers laminar flow in a cylindrical pipe. Here the state variable in question is the axial velocity v, which rises from a value of zero at the wall to a maximum at the centerline before dropping back to zero at the other end of the diameter. Here, again, symmetry considerations dictate that this maximum must be located at the centerline of the conduit. 

The problem of pore diffusion is only limited to immobilized enzyme catalysts, and not enzyme catalyzed reactions in which the enzyme is used in the native or soluble form. Immobilized enzymes are supported catalysts in which the enzyme is supported or immobilized on a suitable inert support such as alumina, kiesulguhr, silica, or microencapsulated in a suitable polymer matrix. The shape of the immobilized enzyme pellet may be spherical, cylindrical, or rectangular (as in a slab). If the reaction follows Michaelis-Menten kinetics discussed previously, then a shell balance around a spherical enzyme pellet results in the following second order differential equation ... 

Examples of the application of Eq. (3.3.17) to spherical particles include the work of Carter [9] for the oxidation of nickel, Kawasaki et al. [10] for the reduction of iron oxides, and Weisz and Goodwin [11] for the combustion of coke deposits on catalysts. In the latter two cases, the solid was initially a porous pellet when diffusion controls the overall rate, however, the relationships above may be used. We will discuss this in more detail when the system of a porous reactant solid is presented. Many studies on the oxidation of metals have been made in one-dimensional geometries [12,13]—hence the term parabolic law for the rate of progress of such oxidation reactions under conditions of diffusion control [see Eq. (3.3.14)]. Hutchins [14] has verified Eq. (3.3.16) experimentally using a cylindrical system. 

---