Split boundary conditions

Data evaluation The evaluation of model parameters by non-linear fitting of experimental net diffusion flux densities to theory requires solution of a set of coupled ordinary differential equations which describe diffusion in porous solids according to MTPM (integration of differential equations with splitted boundary conditions). 

Equation (6.79) is a second order differential equation of the boundary value type having two split boundary conditions at (u = 0... 

X = 0, Trx = Trx, inlet) are available at z = 0. This is known classically as a split boundary value problem, and it is characteristic of countercurrent flow heat exchangers. When numerical methods are required to integrate coupled mass and thermal energy balances subjected to split boundary conditions, it is necessary to do the following ... 

This second-order ODE for 4 a(9) with split boundary conditions, given by equations (19-11) and (19-12), cannot be solved numerically until one invokes stoichiometry and the mass balance with diffusion and chemical reaction to relate the molar densities of aU gas-phase species within the pores of the catalytic pellet... 

The heterogeneous rate law in (22-57) is dimensionalized with pseudo-volumetric nth-order kinetic rate constant k that has units of (volume/mol)" per time. k is typically obtained from equation (22-9) via surface science studies on porous catalysts that are not necessarily packed in a reactor with void space given by interpellet. Obviously, when axial dispersion (i.e., diffusion) is included in the mass balance, one must solve a second-order ODE instead of a first-order differential equation. Second-order chemical kinetics are responsible for the fact that the mass balance is nonlinear. To complicate matters further from the viewpoint of obtaining a numerical solution, one must solve a second-order ODE with split boundary conditions. By definition at the inlet to the plug-flow reactor, I a = 1 at = 0 via equation (22-58). The second boundary condition is d I A/df 0 as 1. This is known classically as the Danckwerts boundary condition in the exit stream (Danckwerts, 1953). For a closed-closed tubular reactor with no axial dispersion or radial variations in molar density upstream and downstream from the packed section of catalytic pellets, Bischoff (1961) has proved rigorously that the Danckwerts boundary condition at the reactor inlet is... 

The numerical solution of second-order nonlinear ODEs with split boundary conditions requires trial and error integration of two coupled first-order ODEs. If one defines d p./di = Axial Grad, then the one-dimensional plug-flow mass balance with axial dispersion,... 

Results from the previous section in this chapter illustrate how and when interpellet axial dispersion plays an important role in the design of packed catalytic tubular reactors. When diffusion is important, more sophisticated numerical techniques are required to solve second-order ODEs with split boundary conditions to predict non-ideal reactor performance. Tubular reactor performance is nonideal when the mass transfer Peclet number is small enough such that interpellet axial dispersion cannot be neglected. The objectives of this section are to understand the correlations for effective axial dispersion coefficients in packed beds and porous media and calculate the mass transfer Peclet number based on axial dispersion. Before one can make predictions about the ideal vs. non-ideal performance of tubular reactors, steady-state mass balances with and without axial dispersion must be solved and the reactant concentration profiles from both solutions must be compared. If the difference between these profiles with and without interpellet axial dispersion is indistinguishable, then the reactor operates ideally. 

It is necessary to solve two coupled first-order ODEs with split boundary conditions to estimate the effect of axial dispersion on outlet conversion. 

Two coupled ODEs are solved with first-order irreversible chemical kinetics, subject to the following split boundary conditions ... 

Answer Two. The thermal energy balance is not required when the enthalpy change for each chemical reaction is negligible, which causes the thermal energy generation parameters to tend toward zero. Hence, one calculates the molar density profile for reactant A within the catalyst via the mass transfer equation, which includes one-dimensional diffnsion and multiple chemical reactions. Stoichiometry is not required because the kinetic rate law for each reaction depends only on Ca. Since the microscopic mass balance is a second-order ordinary differential eqnation, it can be rewritten as two coupled first-order ODEs with split boundary conditions for Ca and its radial gradient. 

This system of equations can be reduced to four first-order linked linear ordinary differential equations with split boundary conditions. The transition point where the two-phase fluid is completely vaporized is unknown. Furthermore, the wall thickness is discontinuous, which means that is also discontinuous. 

Boundary conditions for the above equation, applied to both phases, depend on the specific details of the model chosen. For the simplest case of plug flow in each phase (I gi= g2 upflow in each phase, it is sufficient to set c =C2=Cq at x=0, where Cq is the initial concentration. When downflow occurs in one of the phases, a split boundary condition problem arises. 

So far, only a single reaction has been considered. While the reactor point effectiveness cannot be expressed explicitly for a reversible reaction, the internal effectiveness factor can readily be obtained analytically using the generalized modulus (see Problem 4.23). For complex multiple reactions, however, it is not possible to obtain analytical expressions for the global rates and one has to solve the conservation equations numerically. The numerical solution of nonlinear, coupled diffusion equations with split boundary conditions is by no means trivial and often presents convergence difficulties. In this section, the same approach is taken as was used for the reactor point effectiveness. This enables the global rates to be obtained in a straightforward manner and the diffusion equations to be solved as an initial value problem (Akella 1983). 

Table 10.1. While the design procedures for isothermal reactors and adiabatic reactors are straightforward to use, those for nonadiabatic reactors are complicated by the split boundary conditions on temperature. Procedures for nonadiabatic reactors are summarized in Figure 10.1 in the form of a flow chart. Note that q. (D) in Table 10.1 is the Euler version of q. 10.18 for numerical integration. Using the procedures given in Figure 10.1, a table of r versus Cout can be generated, from which the value of t corresponding to the desired conversion can be selected.

Even with the simplifications made possible using the reactor point effectiveness, the design and analysis problem is still not simple. It should be clear, however, that the alternative of solving nonlinear differential equations with split boundary conditions on each is much less attractive. 

---