Kinetic rate law dimensionless

The most important characteristic of this problem is that the Hougen-Watson kinetic model contains molar densities of more than one reactive species. A similar problem arises if 5 mPappl Hw = 2CaCb because it is necessary to relate the molar densities of reactants A and B via stoichiometry and the mass balance with diffusion and chemical reaction. When adsorption terms appear in the denominator of the rate law, one must use stoichiometry and the mass balance to relate molar densities of reactants and products to the molar density of key reactant A. The actual form of the Hougen-Watson model depends on details of the Langmuir-Hinshelwood-type mechanism and the rate-limiting step. For example, consider the following mechanism  

Each of the other gases chemisorbs on a single active site within the internal pores of the catalytic pellet, and all the adsorption/desorption steps have equilibrated. 

Triple-site chemical reaction on the catalytic surface is rate limiting. 

The heterogeneous rate law with units of moles per area per time can be expressed in terms of partial pressures of each component as follows  

If the dimensionless adsorption terms in the denominator of IRhw are not accounted for, then the reaction is irreversible and second-order. Hence, n = 2 and 

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The dimensionless kinetic rate law in the mass balance is defined by... 

The dimensionless kinetic rate law, which includes adsorption terms in the denominator, is expressed as... 

As expected, a shorter reactor is required to achieve the same final conversion when the characteristic chemical reaction time constant u> is smaller and the effectiveness factor E is larger. Since the integral in equation (22-27) that contains the dimensionless kinetic rate law reduces to a constant when the final conversion of... 

Before one can obtain a numerical value for t from the integral form of the plug-flow reactor design equation, given by (22-27), it is necessary to focus on the dimensionless kinetic rate law, which could be rather complex. 

In calculating most of the reaction paths in this book, we have measured reaction progress with respect to the dimensionless variable . We showed in Chapter 16, however, that by incorporating kinetic rate laws into a reaction model, we can trace reaction paths describing mineral precipitation and dissolution using time as the reaction coordinate. 

Dalton s law is used to express partial pressures in terms of mole fractions and total pressure. Total pressure must be expressed in atmospheres if equilibrium, p IS Calculated via the dimensionless thermodynamic equation for / equilibrium, /, which is based on standard-state enthalpies and entropies of reaction at 298 K. Furthermore, the dimensions of / equilibrium, p are the same as those of / s°tondard state P (atm) for this problem]. The latter equilibrium constant, which is based on standard-state fugacities of pure components at 1 atm and 298 K, has a magnitude of 1 when total pressure in the kinetic rate law is expressed in atmospheres. Hence,... 

Consider the following nine examples of diffusion and chemical reaction in porous catalysts where the irreversible kinetic rate law is only a function of the molar density of reactant A. Identify the problems tabulated below that yield analytical solutions for (a) the molar density of reactant A, and (b) the dimensionless correlation between the effectiveness factor and the intrapellet Damkohler number. 

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. 

Step 1. Use bulk conditions at the reactor inlet, CA,buik gas(z = 0) and Tbuik gas (z = 0), to estimate the intrapellet Damkohler number intrapeiiet responding effectiveness factor E via dimensionless correlations that account for catalyst geometry and the appropriate kinetic rate law (i.e., nth-order kinetics). 

If these equations are used to substitute the individual concentrations in the differ-.ent formal kinetic rate laws, it becomes possible in principle to split the resulting rate laws into one term repres ting the initial rate ro and a second, which stands for a dimensionless reaction rate... 

By applying Equ. 4-232 and using the parameters identified so far, all discrete values of the dimensionless reaction rate 0(X) can be calculated for each individual conversion data point. In a last step the data set couples (X) and X have to be fitted to a formal kinetic rate law. It is recommendable to begin this fitting procedure with the most simple model, the power law, first. 

The dimensionless time (t), potential ( ), and current (i/0 are all as defined in equations (1.4). The exact characteristics of the voltammograms depend on the rate law. In the case of Butler-Volmer kinetics,... 

As a typical example of this type of reaction, the transformation A) — products may be considered, where the kinetics are described by a simple power rate law of the order n. Since this reaction is completely characterized by specifying the conversion of reactant A. the above system of diflcntial equations (eqs 11-18) may be readily expressed in a convenient, nondimcnsional form. For this purpose, the reactant concentration and the temperature arc related to their corresponding values in the bulk fluid phase (eqs 24 and 25), and the radius coordinate r is divided by the pellet radius R to introduce a dimensionless coordinate (eq 26). 

Under isothermal conditions, the kinetic rate constant is truly constant and the dimensionless rate law 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. 

The correction factor Nsur ce in the E vs. Aa correlation for complex kinetics is given by the inverse of the dimensionless rate law evaluated at the external surface of the catalyst, where the dimensionless molar density of reactant A is unity, by definition. Hence, the correction factor surface for the Hougen-Watson model described by equations (19-1) and (19-8) is ... 

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... 

Consider the Hougen-Watson kinetic model for the production of methanol from CO and H2, given by equation (22-38). Do not linearize the rate expression. Write the rate law in dimensionless form if the chemical reaction is essentially irreversible (i.e., A eq, 00). 

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. 

In Chapter 10, the dimensionless scaling factor in the mass transfer equation with diffusion and chemical reaction was written with subscript j for the jth chemical reaction in a multiple-reaction sequence (see equation 10-10). In the absence of convective mass transfer, the number of dimensionless scaling factors in the mass transfer equation for component i is equal to the number of chemical reactions. Hence, corresponds to the Damkohler number for reaction j. The only distinguishing factor between aU of these Damkohler numbers for multiple reactions is that the nth-order kinetic rate constant in the jth reaction (i.e., kj), for a volumetric rate law based on molar densities, changes from one reaction to another. The characteristic length L, the molar density of key-limiting reactant A on the external surface of the catalyst CA.sur ce, and the effective diffusion coefficient of reactant A, a. effective, are the same in aU Damkohler numbers that appear in the dimensionless mass balance for reactant A. In other words. 

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... 

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