Damkohler number boundary conditions

From the boundary conditions, show the dimensionless parameters that the critical Damkohler number will depend on. 

Figure 22.3 One-dimensional concentration profiles at steady-state calculated from the diffusion/advec-tion/reaction equation (Eq. 22-7) for different parameter values D (diffii-sivity), x (advection velocity), and kr (first-order reaction rate constant). Boundary conditions at x = 0 and x - L are C0 and CL, respectively. Pe = 7. vx ID is the Peclet Number, Da = Dk/v] is the Damkohler Number. See text for further explanations.

Develop and discuss a set of boundary conditions to solve the Graetz problem. Take particular care with the effects of surface reaction, balancing heterogeneous reaction with mass diffusion from the fluid. A second Damkohler number should emerge in the surface boundary condition,... 

For known values of the parameters in the kinetic equation for a specific reactive mix, it is easy to calculate the dimensionless factors y and v. Then the flow pattern in the mold filling process is completely determined by the dimensionless Da and Gz Numbers and the boundary conditions. The Damkohler Number characterizes the ratio of the rates of chemical reaction and convective heat transfer and the Graetz Number is a measure of the ratio of the convective heat flux due to a moving liquid to the heat flux due to the conductivity of the liquid. 

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... 

Here Xi and X2 are the dimensionless concentrations and temperatures. Da is the Damkohler number andy the dimensionless longitudinal coordinate. All the parameters are described in [5]. The boundary conditions are ... 

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... 

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,... 

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 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. 

Non-ideal simulations satisfy the Danckweits boundary condition for the outlet concentration gradient. Real and ideal tubular reactor performance at various mass transfer Peclet numbers is compared when the product of the effectiveness factor, the interpeilet Damkohler number, and the catalyst filling factor is 5. 

The boundary conditions at the external surface of the catalyst are T = Tsurface and Ca = Ca surface, and A effeciive is the effective thermal conductivity of the composite catalyst structure (i.e., 1.6 x 10 J/cm s K for alumina). Initially, the surface temperature and concentration of reactant A in Uie vicinity of a single isolated catalytic peUet are chosen to match the inlet values to the packed reactor. If external mass and heat transfer resistances are minimal, then bulk gas-phase temperature and reactant concentration at each axial position in the reactor represent the characteristic quantities that should be used to calculate the intrapellet Damkohler number for nth-order chemical kinetics ... 

The concentration profile as estimated by Equations 8.34 is plotted in Figure 8.4, along with the numerical solution of Equation 8.16, subject to the boundary condition (8.20). The axial variation of the mixing-cup concentration (expressed in terms of the dimensionless Graetz parameter) is shown for several values of the Damkohler number (covering kinetic and mass transfer control). 

In order to obtain the (mixing-cup and surface) concentration and temperature profiles, the solution of Equations 8.52 and 8.55 requires the calculation of the dimensionless transfer coefficients Sh and Nw, which are in general functions of the Graetz parameters and of the Damkohler number. Correlations for these dimensionless numbers are available in the literature (e.g., Shah and London [39]), for several conditions regarding the degree of profile development, degree of hydrodynamic flow development, and boundary condition at the wall. These features are briefly discussed in the following. 

From the above nondimensionalized boundary condition, we can define the dimensionless number, Damkohler number, as... 

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