Coupled transport equations

The appropriate mathematical model for solute transport originating from the dissolution of a perchloroethylene (PCE) pool located onto a bedrock within a two-dimensional, homogeneous, water saturated aquifer in the presence of dissolved humic substances, as illustrated in Fig. 8, consists of three coupled transport equations. One equation describing the transport of the solute originating from the dissolving PCE pool in the presence of dissolved humic substances, another equation describing the transport of dissolved humic substances, and the third equation describing the transport of solute-humic particles. 

Ilias and Govind [1993] also used the CFD approach to solve coupled transport equations of momentum and species describing the dynamics of a tubular ultraflltration or reverse osmosis unit. An implicit finite-difference method was used as the solution scheme. Local variations of solute concentration, u ansmembranc flux and axial pressure drop can be obtained from the simulation which, when compared to published experimental data, shows that the common practice of using a constant membrane permeability (usually obtained from the data of pure water flux) can grossly overestimate... 

For reactive multi-component transport, the main problem is the coupling between components through the reactive term. Indeed the reaction kinetics depends on the concentrations of several components, which makes necessary to consider large system of coupled transport equations. For the case of an irreversible reaction, the situation is simplified because the reaction kinetics depends only on reagents and does not depend on the reaction products. Then it is sufficient to formulate the transport equations only for hydrogen and CC>2. 

Electrochemical charge storage in a bulk film of material consists of the movement of two types of species, into or out of the film. These may be, for example, a cation and an electron. The requirement of charge neutrality stipulates that the two species move essentially together. We may therefore solve the coupled transport equations to give an equation that describes the flux of the neutral combination of species, in terms of the gradient in concentration (36). This allows us to define the chemical diffusion coefficient, which describes the motion of a neutral species in a non-ideal solid solution. 

Simple chemical systems with several components (HCl, KOH, KCl in hydrogel) were used for modeling mass and charge balances coupled with equations for electric field, transport processes and equilibrium reactions [146]. This served for demonstrating the chemical systems function as electrolyte diodes and transistors, so-called electrolyte-microelectronics . 

Especially for the electrons, the fluid model has the advantage of a lower computational effort than the PIC/MC method. Their low mass (high values of the transport coefficients) and consequent high velocities give rise to small time steps in the numerical simulation (uAf < Aa) if a so-called explicit method is used. This restriction is easily eliminated within the fluid model by use of an implicit method. Also, the electron density is strongly coupled with the electric field, which results in numerical Instabilities. This requires a simultaneous implicit solution of the Poisson equation for the electric field and the transport equation for the electron density. This solution can be deployed within the fluid model and gives a considerable reduction of computational effort as compared to a nonsi-multaneous solution procedure [179]. Within the PIC method, only fully explicit methods can be applied. 

These convective transport equations for heat and species have a similar structure as the NS equations and therefore can easily be solved by the same solver simultaneously with the velocity field. As a matter of fact, they are much simpler to solve than the NS equations since they are linear and do not involve the solution of a pressure term via the continuity equation. In addition, the usual assumption is that spatial or temporal variations in species concentration and temperature do not affect the turbulent-flow field (another example of oneway coupling). 

The Eulerian gas velocity field required in both the mass balance and the above transport equation for nh is found by an approximate method first, the complete field of liquid velocities obtained with FLUENT is adapted downward because the power draw is smaller under gassed conditions next, in a very simple way of one-way coupling, the bubble velocity calculated from the above force balance is just added to this adapted liquid velocity field. This procedure makes a momentum balance for the bubble phase redundant this saves a lot of computational effort. 

Mass transport models for multicomponent systems have been developed where the equilibrium interaction chemistry is solved independently of the mass transport equations which leads to a set of algebraic equations for the chemistry coupled to a set of differential equations for the mass transport. (Cederberg et al., 1985). 

A fundamental fuel cell model consists of five principles of conservation mass, momentum, species, charge, and thermal energy. These transport equations are then coupled with electrochemical processes through source terms to describe reaction kinetics and electro-osmotic drag in the polymer electrolyte. Such convection—diffusion—source equations can be summarized in the following general form... 

Six coupled governing equations listed in Table 1 are valid in all regions of a PEFC, and fluxes at an internal boundary between two adjacent regions are automatically continuous. Such a single-domain model is well suited for CFD implementation. In contrast, multidomain models, such as the one developed by Dutta et al., compute separate solutions for the anode and cathode subdomains, respectively, and then patch the two solutions through the water transport flux across the MEA interface. Numerically, this model is characterized as a solver-in-solver situation. 

The derivation of the two-box model follows naturally from the one-box model. It is useful for describing systems consisting of two spatial subsystems which are connected by one or several transport processes. The mass balance equations for the individual boxes look like Eq. 21-1 with the addition of terms describing mass fluxes between the boxes. Each box can be characterized by one or several state variables. Thus, the dimension of the system of coupled differential equations is the product of the number of boxes and the number of variables per box. 

Current densities in the cathode are mainly determined by the respective value of oxide anion conductivity compared to the electronic conductivity (/Co" and ice", coupled to each other in Wagner diffusion). Equation (34) describes the current density limit for coupled transport of oxygen anions and electrons (777) ... 

The system of equations in the Von Mises form leads to a coupled system of nonlinear differential-algebraic equations. The transport equations (Eqs. 7.59 and 7.62) have parabolic characteristics, with the axial coordinate z being the timelike direction. The other three equations (Eqs. 7.60, 7.61, and 7.63) are viewed as algebraic constraints—in the sense that they have no timelike derivatives. 

The goal of this chapter is to describe the application of hydrodynamic electrodes to the study of electrode kinetics and the kinetics of electrode and coupled homogeneous reactions. In order to do this, it is important to describe first the mass transport and how to fulfil experimentally the conditions described by the mass transport equations, i.e. electrode construction and operation. 

However, the calculation of NAO( f)> which is the matrix composition at explicit solution to the coupled diffusion equations of the components before and behind the reaction front. Since the transport coefficients in these mixed crystals depend on local composition, one therefore cannot find analytical solutions. Only if the A2+ ions are almost immobile (DA< DB) do we have NAO( F) =N 0. This specific case has been discussed in the literature [H. Schmalzried (1984)]. 

The reaction scheme at and near the phase boundary during the phase transformation is depicted in Figure 10-14. The width of the defect relaxation zone around the moving boundary is AifR, it designates the region in which the relaxation processes take place. The boundary moves with velocity ub(f) and establishes the boundary conditions for diffusion in the adjacent phases a and p. The conservation of mass couples the various processes. This is shown schematically in Figure 10-14b where the thermodynamic conditions illustrated in Figure 10-12 are also taken into account. The transport equations (Fick s second laws) have to be solved in both the a and p... 

Finally, let us briefly point out some essential features of the stability analysis for a more general transport problem. It can be exemplified by the moving a//9 phase boundary in the ternary system of Figure 11-12. Referring to Figure 11-7 and Eqn. (11.10), it was a single independent (vacancy) flux that caused the motion of the boundary. In the case of two or more independent components, we have to formulate the transport equation (Fick s second law) for each component, both in the a- and /9-phase. Each of the fluxes jf couples at the boundary b with jf, i = A,B,... (see, for example, Eqn. (11.2)). Furthermore, in the bulk, the fluxes are also coupled (e.g., by electroneutrality or site conservation). 

---