Catalytic pellets kinetics

Step 1. Reactants enter a packed catalytic tubular reactor, and they must diffuse from the bulk fluid phase to the external surface of the solid catalyst. If external mass transfer limitations provide the dominant resistance in this sequence of diffusion, adsorption, and chemical reaction, then diffusion from the bulk fluid phase to the external surface of the catalyst is the slowest step in the overall process. Since rates of interphase mass transfer are expressed as a product of a mass transfer coefficient and a concentration driving force, the apparent rate at which reactants are converted to products follows a first-order process even though the true kinetics may not be described by a first-order rate expression. Hence, diffusion acts as an intruder and falsifies the true kinetics. The chemical kineticist seeks to minimize external and internal diffusional limitations in catalytic pellets and to extract kinetic information that is not camouflaged by rates of mass transfer. The reactor design engineer must identify the rate-limiting step that governs the reactant product conversion rate. 

Most important, heterogeneous surface-catalyzed chemical reaction rates are written in pseudo-homogeneous (i.e., volumetric) form and they are included in the mass transfer equation instead of the boundary conditions. Details of the porosity and tortuosity of a catalytic pellet are included in the effective diffusion coefficient used to calculate the intrapellet Damkohler number. The parameters (i.e., internal surface area per unit mass of catalyst) and Papp (i.e., apparent pellet density, which includes the internal void volume), whose product has units of inverse length, allow one to express the kinetic rate laws in pseudo-volumetric form, as required by the mass transfer equation. Hence, the mass balance for homogeneous diffusion and multiple pseudo-volumetric chemical reactions in one catalytic pellet is... 

In other words, reactants exist everywhere within the pores of the catalyst when the chemical reaction rate is slow enough relative to intrapellet diffusion, and the intrapellet Damkohler number is less than, or equal to, its critical value. These conditions lead to an effectiveness factor of unity for zerofli-order kinetics. When the intrapellet Damkohler number is greater than Acnticai, the central core of the catalyst is reactant starved because criticai is between 0 and 1, and the effectiveness factor decreases below unity because only the outer shell of the pellet is used to convert reactants to products. In fact, the dimensionless correlation between the effectiveness factor and the intrapeUet Damkohler number for zeroth-order kinetics exhibits an abrupt change in slope when A = Acriticai- Critical spatial coordinates and critical intrapeUet Damkohler numbers are not required to analyze homogeneous diffusion and chemical reaction problems in catalytic pellets when the reaction order is different from zeroth-order. When the molar density appears explicitly in the rate law for nth-order chemical kinetics (i.e., n > 0), the rate of reaction antomaticaUy becomes extremely small when the reactants vanish. Furthermore, the dimensionless correlation between the effectiveness factor and the intrapeUet Damkohler nnmber does not exhibit an abrupt change in slope when the rate of reaction is different from zeroth-order. 

Reactant concentrations and effectiveness factors are calculated for diffusion and second-order kinetics in isothermal catalytic pellets via the methodology described above. In each case, convergence is achieved when the dimensionless molar density of reactant A at the external surface is 1 5, where S is on the order of 10 or less. 

NUMERICAL SOLUTIONS FOR DIFFUSION AND HOUGEN-WATSON CHEMICAL KINETICS IN ISOTHERMAL CATALYTIC PELLETS... 

The simplified homogeneous mass transfer model for diffusion and Langmuir- Hin-shelwood chemical kinetics within the internal pores of an isolated catalytic pellet is written in dimensionless form for reactant A or A2 (i.e., ua = —1) ... 

When there are j chemical reactions between i species in a mixture, it is possible to constract a Damkohler number for reaction j that is specific to component i. This is necessary because the effective pore diffusion coefficient within a catalytic pellet depends on molecular size. Hence, if reaction j is described by nth-order irreversible chemical kinetics, then the Damkohler number of component i in the reactive gas mixture is... 

Notice that the molar density of key-limiting reactant A on the external surface of the catalytic pellet is always used as the characteristic quantity to make the molar density of component i dimensionless in all the species mass balances. effective is the effective intrapellet diffusion coefficient of species i. If there is only one chemical reaction, or one rate-limiting step in a multiple reaction sequence, that is characterized by nth-order irreversible kinetics, then the rate constant in the numerator of the Damkohler numbers is the same for each A -. Hence, kj is written as k , which signifies that has units of (volume/mole)" /time for... 

The mass balance with homogeneous one-dimensional diffusion and irreversible nth-order chemical reaction provides basic information for the spatial dependence of reactant molar density within a catalytic pellet. Since this problem is based on one isolated pellet, the molar density profile can be obtained for any type of chemical kinetics. Of course, analytical solutions are available only when the rate law conforms to simple zeroth- or first-order kinetics. Numerical techniques are required to solve the mass balance when the kinetics are more complex. The rationale for developing a correlation between the effectiveness factor and intrapellet Damkohler number is based on the fact that the reactor design engineer does not want to consider details of the interplay between diffusion and chemical reaction in each catalytic pellet when these pellets are packed in a large-scale reactor. The strategy is formulated as follows ... 

Since is only a function of spatial coordinate r), the partial derivative in equation (20-7) is replaced by a total derivative, and the dimensionless concentration gradient evaluated at the external surface (i.e = 1) is a constant that can be removed from the surface integral in the numerator of the effectiveness factor (see equation 20-6). For simple nth-order irreversible chemical kinetics in catalytic pellets, where the rate law is a function of the molar density of only one reactant. 

It is rather straightforward to employ numerical methods and demonstrate that the effectiveness factor approaches unity in the reaction-rate-controlled regime, where A approaches zero. Analytical proof of this claim for first-order irreversible chemical kinetics in spherical catalysts requires algebraic manipulation of equation (20-57) and three applications of rHopital s rule to verify this universal trend for isothermal conditions in catalytic pellets of any shape. 

Notice that the molar density of key-limiting reactant A on the external surface of the catalytic pellet is always used as the characteristic quantity to make the molar density of component i dimensionless in all the component mass balances. This chapter focuses on explicit numerical calculations for the effective diffusion coefficient of species i within the internal pores of a catalytic pellet. This information is required before one can evaluate the intrapellet Damkohler number and calculate a numerical value for the effectiveness factor. Hence, 50, effective is called the effective intrapellet diffusion coefficient for species i. When 50, effective appears in the denominator of Ajj, the dimensionless scaling factor is called the intrapellet Damkohler number for species i in reaction j. When the reactor design focuses on the entire packed catalytic tubular reactor in Chapter 22, it will be necessary to calcnlate interpellet axial dispersion coefficients and interpellet Damkohler nnmbers. When there is only one chemical reaction that is characterized by nth-order irreversible kinetics and subscript j is not required, the rate constant in the nnmerator of equation (21-2) is written as instead of kj, which signifies that k has nnits of (volume/mole)"" per time for pseudo-volumetric kinetics. Recall from equation (19-6) on page 493 that second-order kinetic rate constants for a volnmetric rate law based on molar densities in the gas phase adjacent to the internal catalytic surface can be written as... 

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 effectiveness factor E is evaluated for the appropriate kinetic rate law and catalyst geometry at the corresponding value of the intrapellet Damkohler number of reactant A. When the resistance to mass transfer within the boundary layer external to the catalytic pellet is very small relative to intrapellet resistances, the dimensionless molar density of component i near the external surface of the catalyst (4, surface) IS Very similar to the dimensionless molar density of component i in the bulk gas stream that moves through the reactor ( I, ). Under these conditions, the kinetic rate law is evaluated at bulk gas-phase molar densities, 4, . This is convenient because the convective mass transfer term on the left side of the plug-flow differential design equation d p /di ) is based on the bulk gas-phase molar density of reactant A. The one-dimensional mass transfer equation which includes the effectiveness factor. 

When the kinetics are first-order and irreversible in catalytic pellets with spherical symmetry, the mass transfer/chemical reaction model that focuses on intrapeUet diffusion is written in dimensionless form for carbon monoxide as... 

The heterogeneous rate law in (22-57) is dimensionalized with pseudo-volumetric nth-order kinetic rate constant k that has units of (volume/mol)" per time. k is typically obtained from equation (22-9) via surface science studies on porous catalysts that are not necessarily packed in a reactor with void space given by interpellet. Obviously, when axial dispersion (i.e., diffusion) is included in the mass balance, one must solve a second-order ODE instead of a first-order differential equation. Second-order chemical kinetics are responsible for the fact that the mass balance is nonlinear. To complicate matters further from the viewpoint of obtaining a numerical solution, one must solve a second-order ODE with split boundary conditions. By definition at the inlet to the plug-flow reactor, I a = 1 at = 0 via equation (22-58). The second boundary condition is d I A/df 0 as 1. This is known classically as the Danckwerts boundary condition in the exit stream (Danckwerts, 1953). For a closed-closed tubular reactor with no axial dispersion or radial variations in molar density upstream and downstream from the packed section of catalytic pellets, Bischoff (1961) has proved rigorously that the Danckwerts boundary condition at the reactor inlet is... 

The analysis in this section focuses on the appropriate dimensionless numbers that are required to analyze convection, axial dispersion and first-order irreversible chemical reaction in a packed catalytic tubular reactor. The catalytic pellets are spherical. Hence, an analytical solution for the effectiveness factor is employed, based on first-order irreversible chemical kinetics in catalysts with spherical symmetry. It is assumed that the catalytic pores are larger than 1 p.m (i.e., > 10 A) and that the operating pressure is at least 1 atm. Under these conditions, ordinary molecular diffusion provides the dominant resistance to mass transfer within the pores because the Knudsen diffusivity,... 

At high-mass-transfer Peclet numbers, sketch the relation between average residence time divided by the chemical reaction time constant (i.e., r/co) for a packed catalytic tubular reactor versus the intrapeUet Damkohler number Aa, intrapeiiet for zeroth-, first-, and second-order irreversible chemical kinetics within spherical catalytic pellets. The characteristic length L in the definition of Aa, intrapeiiet is the sphere radius R. The overall objective is to achieve the same conversion in the exit stream for all three kinetic rate laws. Put all three curves on the same set of axes and identify quantitative values for the intrapeiiet Damkohler number on the horizontal axis. 

Inlet gas-phase molar density of reactant A, Cao = 3 x 10 g-mol/cm First-order kinetic rate constant for the surface-catalyzed chemical reaction based on gas-phase molar densities, surface = 5 x 10 cm/min (also known as the reaction velocity constant) — this is not a pseudo-volumetric kinetic rate constant Diameter of spherically shaped catalytic pellets = 1 cm Intrapeiiet porosity factor = 65% (i.e., 0.65)... 

For a given length Lpfr of a tubular reactor packed with spherical catalytic pellets, one calculates the following values of five important dimensionless design parameters. The chemical kinetics are first-order and irreversible ... 

Consider one-dimensional (i.e., radial) diffusion and multiple chemical reactions in a porous catalytic pellet with spherical symmetry. For each chemical reaction, the kinetic rate law is given by a simple nth-order expression that depends only on the molar density of reactant A. Furthermore, the thermal energy generation parameter for each chemical reaction, Pj = 0. 

Sketch the molar density of reactant A on the external surface of porous catalytic pellets (Ca, surface) as a function of reactor volume (Vpfr) for an ideal PFR with significant external mass transfer resistance when the chemical kinetics are ... 

The macro-scale physical character of catalysts refers to the characteristics of volume, shape and size distribution as well as related mechanical strength formed by the size, shape and void structure of particles and pellets. Industrial catalyst should have good macro-scale physical character, including surface area, pore volume, pore size and distribution, packing density, favorable particle size and shape and good mechanical strength. These properties not only influence the behavior of mass transfer, heat transfer and hydrodynamics (three transferee), but also directly influence the process of catalytic reaction kinetics. Therefore, macro-scale physical behaviours of catalysts is very important in the research of industrial catalyst. 

Diffusion effects can be expected in reactions that are very rapid. A great deal of effort has been made to shorten the diffusion path, which increases the efficiency of the catalysts. Pellets are made with all the active ingredients concentrated on a thin peripheral shell and monoliths are made with very thin washcoats containing the noble metals. In order to convert 90% of the CO from the inlet stream at a residence time of no more than 0.01 sec, one needs a first-order kinetic rate constant of about 230 sec-1. When the catalytic activity is distributed uniformly through a porous pellet of 0.15 cm radius with a diffusion coefficient of 0.01 cm2/sec, one obtains a Thiele modulus y> = 22.7. This would yield an effectiveness factor of 0.132 for a spherical geometry, and an apparent kinetic rate constant of 30.3 sec-1 (106). 

Many elements of a mathematical model of the catalytic converter are available in the classical chemical reactor engineering literature. There are also many novel features in the automotive catalytic converter that need further analysis or even new formulations the transient analysis of catalytic beds, the shallow pellet bed, the monolith and the stacked and rolled screens, the negative order kinetics of CO oxidation over platinum,... 

It is possible to eliminate the mass transfer resistances in Steps 2, 3, 7, and 8 by grinding the catalyst to a fine powder and exposing it to a high-velocity gas stream. The concentrations of reactants immediately adjacent to the catalytic surface are then equal to the concentrations in the bulk gas phase. The resulting kinetics are known as intrinsic kinetics since they are intrinsic to the catalyst surface and not to the design of the pores, or the pellets, or the reactor. 

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