Intrapellet Damkohler number catalytic reactor design

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 ... 

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... 

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. 

The effectiveness factor E is expressed in terms of the intrapellet Damkohler number, and the chemical reaction time constant co is the inverse of the best pseudo-first-order kinetic rate constant. The reactor design engineer employs an integral form of the design equation to predict the length of a packed catalytic tubular reactor Lpfr that will achieve a final conversion of CO specified by /final. The approximate analytical solution, vahd at high mass transfer Peclet numbers, is... 

Now, all of the tools required to calculate the molar density of reactant A on the external surface of the catalyst are available to the reactor design engineer. It is important to realize that Ca, surface is the characteristic molar density, or normalization factor, for all molar densities within the catalyst. Hence, Ca, surface only appears in the expression for the intrapellet Damkohler number (i.e., excluding first-order kinetics) when isolated pellets are analyzed. Furthermore, intrapellet Damkohler numbers are chosen systematically to calculate effectiveness factors via numerical analysis of coupled sets of dimensionless differential equations. Needless to say, it was never necessary to obtain numerical values for Ca, sur ce in Part IV of this textbook. Under realistic conditions in a packed catalytic reactor, it is necessary to (1) predict Ca, surface and Tsurface, (2) calculate the intrapellet Damkohler number, (3) estimate the effectiveness factor via correlation, (4) predict the average rate of reactant consumption throughout the catalyst, and (5) solve coupled ODEs to predict changes in temperature and reactant molar density within the bulk gas phase. The complete methodology is as follows ... 

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. 

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