Catalytic Damkohler number

C.J. van Duijn, Andro Mikelic, I.S. Pop, and Carole Rosier, Effective Dispersion Equations for Reactive Flows with Dominant Peclet and Damkohler Numbers Mark Z. Lazman and Gregory S. Yablonsky, Overall Reaction Rate Equation of Single-Route Complex Catalytic Reaction in Terms of Hypergeometric Series A.N. Gorban and O. Radulescu, Dynamic and Static Limitation in Multiscale Reaction Networks, Revisited... 

Analysis and Experimental Investigation of Catalytic Membrane Reactors In this equation, Da is the Damkohler number [14]... 

The effective diffusivity Dn decreases rapidly as carbon number increases. The readsorption rate constant kr n depends on the intrinsic chemistry of the catalytic site and on experimental conditions but not on chain size. The rest of the equation contains only structural catalyst properties pellet size (L), porosity (e), active site density (0), and pore radius (Rp). High values of the Damkohler number lead to transport-enhanced a-olefin readsorption and chain initiation. The structural parameters in the Damkohler number account for two phenomena that control the extent of an intrapellet secondary reaction the intrapellet residence time of a-olefins and the number of readsorption sites (0) that they encounter as they diffuse through a catalyst particle. For example, high site densities can compensate for low catalyst surface areas, small pellets, and large pores by increasing the probability of readsorption even at short residence times. This is the case, for example, for unsupported Ru, Co, and Fe powders. 

In the above Da denotes the Damkohler number as the ratio of the characteristic process time H/V to the characteristic reaction time l/r0. The reaction rate r0 is a reference value at the system pressure and an arbitrary reference temperature, as the lowest or the highest boiling point. For catalytic reactions r0 includes a reference value of the catalyst amount. R is the dimensionless reaction rate R = r/r0. The kinetics of a homogeneous liquid-phase reaction is described in general as function of activities ... 

The catalytic reaction is simply a bimolecular reaction between B and R, with boundary conditions given by ce,m lz=0+ = cxBm, cj >m z=0+ = c] m. The yield of S increases monotonically as the Damkohler number of the catalytic reaction, Das, increases, and finally attains an asymptotic value when the catalytic reaction reaches its mass transfer limited asymptote. This feature is illustrated in Fig. 19, where the variation of Ys with Das is shown. It is interesting to note from Fig. 19, that the value of the mass transfer limited asymptote depends on the micromixing limitation of the homogeneous reaction. Larger is the micromixing limitation (rj) of the homogeneous reaction, more is the local... 

Das reactor scale Damkohler number (catalytic reaction)... 

IS a modified Damkohler number = A nhsCno ts the dimensionless NH3 adsoiption constant, D, is the molecular diffusivity of species 1 is the effective intraporous diffusivity of species i evaluated according to the Wakao-Smith random pore model [411. Equation (4) is taken from Ref. 39. Equations (6)-(8) provide an approximate analytical solution of the intraporous diffusion-reaction equations under the assumption of large Thiele moduli (i.e., the concentration of the limiting reactant is zero at the centerline of the catalytic wall) the same equations are solved numencally in Ref. 36. 

Obviously, the Damkohler numbers are important when chemical reaction occurs, as illustrated by these two examples, which include diffusion and pseudo-homogeneous chemical reaction in porous catalytic pellets. Details of the diffusion equation without convection in nonreactive systems are summarized below for transient and steady-state analyses ... 

Step 2. Reactants must diffuse into the central core of the porous catalyst. A quantitative description of this diffusion process requires knowledge of the tortuosity factor of the pellet, which accounts for the tortuous pathway that strongly influences diffusion. The reactor design engineer seeks numerical values for the inflapellet Damkohler number and the effectiveness factor to characterize intrapellet diffusion in an isolated catalytic pellet. 

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. 

When the intrapellet Damkohler number is less than its critical value (i.e., /6), the critical dimensionless spatial coordinate Jjcnticai is negative, and boundary condition 2b must be employed instead of 2a. Under these conditions, the dimensionless molar density profile for reactant A within the catalytic pores is adopted from equation (16-24) by setting //criucai to zero. Hence,... 

Hence, it is not possible to redefine the characteristic length such that the critical value of the intrapellet Damkohler number is the same for all catalyst geometries when the kinetics can be described by a zeroth-order rate law. However, if the characteristic length scale is chosen to be V cataiyst/ extemai, then the effectiveness factor is approximately A for any catalyst shape and rate law in the diffusion-limited regime (A oo). This claim is based on the fact that reactants don t penetrate very deeply into the catalytic pores at large intrapellet Damkohler numbers and the mass transfer/chemical reaction problem is well described by a boundary layer solution in a very thin region near the external surface. Curvature is not important when reactants exist only in a thin shell near T] = I, and consequently, a locally flat description of the problem is appropriate for any geometry. These comments apply equally well to other types of kinetic rate laws. 

What is the critical value of the intrapeUet Damkohler number for onedimensional diffusion and zeroth-order irreversible chemical reaction in catalytic pellets with spherical symmetry The radius of the sphere is used as the characteristic length in flie definition of the Damkohler number. 

Hence, at the center of spherical catalytic pellets, the first term on the right side of the mass transfer equation with diffusion and chemical reaction depends on the intrapellet Damkohler number and adopts a value between zero and — when the... 

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... 

Reactant equilibrium constants Kp and affect the forward kinetic rate constant, and all Ki s affect die adsorption terms in the denominator of the Hougen-Watson rate law via the 0, parameters defined on page 493. However, the forward kinetic rate constant does not appear explicitly in the dimensionless simulations because it is accounted for in Ihe numerator of the Damkohler number, and is chosen independently to initiate the calculations. Hence, simulations performed at larger adsorption/desorption equilibrium constants and the same intrapellet Damkohler number implicitly require that the forward kinetic rate constant must decrease to offset the increase in reactant equilibrium constants. The vacant-site fraction on the internal catalytic surface decreases when adsorption/desorption equilibrium constants increase. The forward rate of reaction for the triple-site reaction-controlled Langmuir-Hinshelwood mechanism described on page 491 is proportional to the third power of the vacant-site fraction. Consequently, larger T, s at lower temperature decrease the rate of reactant consumption and could produce reaction-controlled conditions. This is evident in Table 19-3, because the... 

Six equations are required to calculate the effectiveness factor for a spherical catalytic pellet when the intrapellet Damkohler number is 10. The elementary irreversible chemical reaction is... 

At relatively low pressures, what dimensionless differential equations must be solved to generate basic information for the effectiveness factor vs. the intrapellet Damkohler number when an isothermal irreversible chemical reaction occurs within the internal pores of flat slab catalysts. Single-site adsorption is reasonable for each component, and dual-site reaction on the catalytic surface is the rate-limiting step for A -h B C -h D. Use the molar density of reactant A near the external surface of the catalytic particles as a characteristic quantity to make all of the molar densities dimensionless. Be sure to define the intrapellet Damkohler number. Include all the boundary conditions required to obtain a unique solution to these ordinary differential equations. 

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 ... 

Conditions 2 and 3 are discussed further in Chapters 22 and 30 that focus on packed catalytic tubular reactors. Condition 1 is addressed by defining the effectiveness factor and using basic information from the mass balance to develop a correlation between the effectiveness factor and the intrapellet Damkohler number. The ratio of reaction rates described below in (a) and (b), with units of moles per time, is defined as the effectiveness factor. 

For a particular experiment in a packed catalytic tubular reactor, the chemical kinetics can be approximated by a zeroth-order rate law where the best value for the zeroth-order rate constant is calculated via the formalism on pages 459 and 460. At what value of the intrapeUet Damkohler number Aa. intrapellet does reactant A occupy 75% by volume of the catalyst if the porous pellets are (a) spherical, (b) long cylinders, and (c) wafer-like ... 

What is the defining expression for the isothermal effectiveness factor in spherical catalysts Reactant A is consumed by three independent first-order irreversible chemical reactions on the interior catalytic surface. Your final expression should be based on mass transfer via diffusion and include the reactant concentration gradient at the external surface of the catalyst, where t) = 1. Define the intrapellet Damkohler number in your final answer. 

Two expressions are given below to calculate the effectiveness factor E. The first one is exact for nth-order irreversible chemical reaction in catalytic pellets, where a is a geometric factor that accounts for shape via the surface-to-volume ratio. The second expression is an approximation at large values of the intrapellet Damkohler number A in the diffusion-limited regime. 

DIFFUSION COEFFICIENTS AND DAMKOHLER NUMBERS WITHIN THE INTERNAL PORES OF CATALYTIC PELLETS... 

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