Molar density profile

In spherical coordinates, the dimensional mass transfer equation with radial diffusion and first-order irreversible chemical reaction exhibits an analytical solution for the molar density profile of reactant A. If the kinetics are not zeroth-order or first-order, then the methodology exists to find the best pseudo-first-order rate constant to match the actual rate law and obtain an approximate analytical solution. The concentration profile of reactant A in the liquid phase must satisfy 

This mass balance with diffusion and chemical reaction is written explicitly in spherical coordinates when reactant A is consumed by one first-order irreversible reaction  

the radial contribution to the spherical-coordinate Laplacian of molar 

Now the mass transfer eqnation for I a(t) with radial diffusion and chemical reaction exhibits a flat description in spherical coordinates  

However, the transformation from Ca to I a via equation (13-4) does not simplify the corresponding mass transfer problem in cylindrical coordinates. The bonndary conditions on hA become 

---

Do any of the transient molar density profiles exhibit overshoot with respect to their steady-state values [Ans. Yes, reactant A in both CSTRs.]... 

If one constructs the appropriate dimensionless equation that governs the molar density profile fi for component i, then xj/i depends on all the dimensionless independent variables and parameters in the governing equation and its supporting boundary conditions. Geometry also plays a role in the final expression for in each case via the coordinate system that best exploits the summetry of the macroscopic boundaries, but this effect is not as important as the dependence of on the dimensionless numbers in the mass transfer equation and its boundary conditions. For example, if convection, diffusion, and chemical reaction are important rate processes that must be considered, then the governing equation for transient analysis... 

Notice that the molar density profiles for these problems are not affected by any dimensionless numbers because either there is only one mass transfer rate process for steady-state analysis, or both rate processes are described by the same dimensional scaling factor. These qualitative trends should be considered before one seeks quantitative information about a particular mass transfer problem. 

Now define a dimensionless concentration variable P (i.e., for the molar density profile) for mobile component A, such that... 

Error Function Molar Density Profile. The fact that conditions (11-198 ) and... 

This allows one to calculate the dimensionless molar density profile RiX) for O2 transport in water at the particular values of r and B mentioned above. Since the gas-liquid interface is characterized by zero shear and perfect slip, P(X) is obtained from the incomplete gamma function when the argument n = and the variable A. = f The first three terms of the infinite series yield the following result ... 

The mass transfer equation for the dimensionless molar density profile of mobile component A is... 

Combination of variables will be successful if the mass transfer equation can be written exclusively in terms of f. For example, if one substitutes the three previous partial derivatives of the dimensionless molar density profile into the mass transfer equation for species A, then the following equation is obtained after multiplication by S ... 

The boundary condition at r] = 0 reveals that D = / bubbieCAi and the fact that 4>a vanishes at the outer edge of the mass transfer boundary layer (i.e., at rj = 1) gives C = —D/tanh A. The molar density profile of reactant A in terms of hyperbolic functions is... 

SOLUTION. The steady-state molar density profile of reactant A, given by (13-14) in the presence of a first-order irreversible chemical reaction, is employed to calculate the r component of the molar flux of A at the solid-liquid interface [i.e., r = (t)]. Then, one constructs an unsteady-state macroscopic mass balance... 

The second integration generates the molar density profile ... 

These critical values are obtained directly from the reactant molar density profile by implementing the condition that = 0 at = / critical- Hence, the critical dimensionless spatial coordinate is calculated by solving for the appropriate root of the following nonlinear algebraic equation ... 

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,... 

Draw the dimensionless molar density profile of reactant A within a porous wafer catalyst for the following values of the intrapeUet Damkohler number. The reaction kinetics are zeroth-order and the characteristic length L is one-half of the wafer thickness, measured in the thinnest dimension. Put all five curves on the same set of axes and be as quantitative as possible on both axes. Dimensionless molar density I a is on the vertical axis and dimensional spatial coordinate rj is on the horizontal axis. 

When the roots of the characteristic equation are real, the molar density profile can be written in terms of hyperbolic sines and cosines. Imaginary roots of the characteristic equation lead to trigonometric sines and cosines. Hence,... 

Hence, the complete solution for the molar density profile within the catalyst is... 

Two-dimensional diffusion occurs axially and radially in cylindrically shaped porous catalysts when the length-to-diameter ratio is 2. Reactant A is consumed on the interior catalytic surface by a Langmuir-Hinshelwood mechanism that is described by a Hougen-Watson kinetic model, similar to the one illustrated by equation (15-26). This rate law is linearized via equation (15-30) and the corresponding simulationpresented in Figure 15-1. Describe the nature of the differential equation (i.e., the mass transfer model) that must be solved to calculate the reactant molar density profile inside the catalyst. 

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 ... 

As illustrated below, the gradient of the dimensionless reactant molar density profile is a function of the intrapeUet Damkohler number, so the effectiveness factor is only a function of A and geometry. Numerical values of a are 1, 2, or 3 for catalysts with rectangular, cyhndrical, or spherical symmetry, respectively. 

The dimensionless molar density profile of reactant A is symmetric with respect to T] about the symmetry plane (i.e., z = 0, r) = 0). Consequently, it is only necessary to integrate equation (20-37) from the symmetry plane at the center of the wafer to the external surface, and multiply by 2. The final expression for the effectiveness factor in rectangular coordinates is... 

Equations (20-48) require knowledge of the dimensionless molar density profile to calculate the molar flux of reactant A into the pellet via Pick s law. At first glance, equations (20-47) allow one to calculate the effectiveness factor for zeroth-order kinetics via trivial integration that does not require knowledge of the molar density profile, because n = 0. Hence,... 

Obviously, the molar density profile is required to calculate the effectiveness factor for zeroth-order kinetics when A > Acnticai because ijciiticai = /(A) is defined by I a = 0. 

Problem. Consider zeroth-order chemical kinetics in pellets with rectangular, cylindrical and spherical symmetry. Dimensionless molar density profiles have been developed in Chapter 16 for each catalyst geometry. Calculate the effectiveness factor when the intrapellet Damkohler number is greater than its critical value by invoking mass transfer of reactant A into the pellet across the external surface. Compare your answers with those given by equations (20-50). 

Mass flux of reactant A into the catalyst across its external surface is employed to develop analytical expressions for the effectiveness factor in terms of the intrapellet Damkohler nnmber. Reactant molar density profiles for diffusion and first-order irreversible reaction have been developed in three coordinate systems, and these profiles in Chapter 17 represent the starting point to calculate the dimensionless concentration gradient on the external surface of the catalyst. In each case, the reader should verify these effectiveness factor results by volumetri-cally averaging the dimensionless molar density profile throughout the pellet via equations (20-47) with n = 1, realizing that it is not necessary to introduce a critical dimensionless spatial coordinate when the kinetics are first-order. 

Long Cylindrical Catalysts. This problem is more difficult because Bessel functions are required to solve the mass transfer equation. The dimensionless molar density profile for reactant A is given by the following classic result (see Section 17-2) ... 

Spherical Catalytic Pellets. Diffusion and first-order chemical reaction in spherical coordinates is a classic chemical engineering problem. Basic information for the dimensionless molar density profile of reactant A is given by (see Section 17-3) ... 

Catalysts with Cylindrical Symmetry. This analysis is based on the mass transfer equation with diffusion and chemical reaction. Basic information has been obtained for the dimensionless molar density profile of reactant A. For zeroth-order kinetics, the molar density is equated to zero at the critical value of the dimensionless radial coordinate, criticai = /(A). The relation between the critical value of the dimensionless radial coordinate and the intrapeUet Damkohler number is obtained by solving the following nonlinear algebraic equation ... 

---