Steady-state diffusion Numerical methods

EXAMPLE 6.6-1. Numerical Method for Convection and Steady-State Diffusion... 

Numerical Method for Steady-State Diffusion. Using the results from Example... 

Numerical Method with Fixed Surface Concentrations. Steady-state diffusion is... 

The nonlinearity of the system of partial differential equations (51) and (52) poses a serious obstacle to finding an analytical solution. A reported analytical solution for the nonlinear problem of diffusion coupled with complexation kinetics was erroneous [12]. Thus, techniques such as the finite element method [53-55] or appropriate change of variables (applicable in some cases of planar diffusion) [56] should be used to find the numerical solution. One particular case of the nonlinear problem where an analytical solution can be given is the steady-state for fully labile complexes (see Section 3.3). However, there is a reasonable assumption for many relevant cases (e.g. for trace elements such as... 

Solution of the coupled mass-transport and reaction problem for arbitrary chemical kinetic rate laws is possible only by numerical methods. The problem is greatly simplified by decoupling the time dependence of mass-transport from that of chemical kinetics the mass-transport solutions rapidly relax to a pseudo steady state in view of the small dimensions of the system (19). The gas-phase diffusion problem may be solved parametrically in terms of the net flux into the drop. In the case of first-order or pseudo-first-order chemical kinetics an analytical solution to the problem of coupled aqueous-phase diffusion and reaction is available (19). These solutions, together with the interfacial boundary condition, specify the concentration profile of the reagent gas. In turn the extent of departure of the reaction rate from that corresponding to saturation may be determined. Finally criteria have been developed (17,19) by which it may be ascertained whether or not there is appreciable (e.g., 10%) limitation to the rate of reaction as a consequence of the finite rate of mass transport. These criteria are listed in Table 1. 

In the second chapter we consider steady-state and transient heat conduction and mass diffusion in quiescent media. The fundamental differential equations for the calculation of temperature fields are derived here. We show how analytical and numerical methods are used in the solution of practical cases. Alongside the Laplace transformation and the classical method of separating the variables, we have also presented an extensive discussion of finite difference methods which are very important in practice. Many of the results found for heat conduction can be transferred to the analogous process of mass diffusion. The mathematical solution formulations are the same for both fields. 

NUMERICAL METHODS FOR STEADY-STATE MOLECULAR DIFFUSION IN TWO DIMENSIONS... 

Derivation of method for steady state. In Fig. 6.6-1 a two-dimensional solid shown with unit thickness is divided into Squares. The numerical methods for steady-state molecular diffusion are very similar to those for steady-state heat conduction discussed in Section 4.15. Hence, only a brief summary will be given here. The solid inside of a square is imagined to be concentrated at the center of the square at c and is called a node, which is connected to the adjacent nodes by connecting rods through which the mass diffuses. 

Sec. 6.6 Numerical Methods for Steady-State Molecular Diffusion in Two Dimensions 413... 

We employ a method of numerical continuation which has been earlier developed into a software tool for analysis of spatiotemporal patterns emerging in systems with simultaneous reaction, diffusion and convection. As an example, we take a catalytic cross-flow tubular reactor with first order exothermic reaction kinetics. The analysis begins with determining stability and bifurcations of steady states and periodic oscillations in the corresponding homogeneous system. This information is then used to infer the existence of travelling waves which occur due to reaction and diffusion. We focus on waves with constant velocity and examine in some detail the effects of convection on the fiiont waves which are associated with bistability in the reaction-diffusion system. A numerical method for accurate location and continuation of front and pulse waves via a boundary value problem for homo/heteroclinic orbits is used to determine variation of the front waves with convection velocity and some other system parameters. We find that two different front waves can coexist and move in opposite directions in the reactor. Also, the waves can be reflected and switched on the boundaries which leads to zig-zag spatiotemporal patterns. 

Derive steady-state and nonsteady-state mass and energy balances for a catalyst monolith channel in which several chemical reactions take place simultaneously. External and internal mass transfer limitations are assumed to prevail. The flow in the chaimel is laminar, but radial diffusion might play a role. Axial heat conduction in the solid material must be accounted for. For the sake of simplicity, use cylindrical geometry. Which numerical methods do you recommend for the solution of the model ... 

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