Spatially lumped models

In the development of molten carbonate fuel cells (MCFCs), many issues require mathematical models. Some of them, for example, the design of controllers and the integration of an MCFC stack in a larger plant system, can be solved with spatially lumped models. Other questions such as the analysis of an inhomogeneous current density profile or the optimal design and operation of a fuel cell with respect to temperature Hmitations, need spatially distributed models. Because the latter are usually the more complex models, this chapter is focused on these models. 

Bosio et al. [45] Spatially lumped model A, C Literature data [7] ... 

Four categories of electrode models can be identified from Table 28.3 the spatially lumped model, the thin-film model, the agglomerate model, and the volume-averaged model. Schemes of the basic concepts of these four model categories are depicted in Figure 28.4. Each of the schemes shows an electrode pore, with the gas channels located at the top and the liquid electrolyte, depicted in gray, at the bottom. In some models, electrolyte is also present in the pore. The reaction zones are indicated by a black face (spot, line, or grid structure). Fluxes of mass and ions are indicated by arrows. 

Calculated reaction rates can be in the spatially ID model corrected using the generalized effectiveness factor (rf) approach for non-linear rate laws. The effect of internal diffusion limitations on the apparent reaction rate Reff is then lumped into the parameter evaluated in dependence on Dc>r, 8 and Rj (cf. Aris, 1975 Froment and Bischoff, 1979, 1990 Leclerc and Schweich, 1993). 

Process-scale models represent the behavior of reaction, separation and mass, heat, and momentum transfer at the process flowsheet level, or for a network of process flowsheets. Whether based on first-principles or empirical relations, the model equations for these systems typically consist of conservation laws (based on mass, heat, and momentum), physical and chemical equilibrium among species and phases, and additional constitutive equations that describe the rates of chemical transformation or transport of mass and energy. These process models are often represented by a collection of individual unit models (the so-called unit operations) that usually correspond to major pieces of process equipment, which, in turn, are captured by device-level models. These unit models are assembled within a process flowsheet that describes the interaction of equipment either for steady state or dynamic behavior. As a result, models can be described by algebraic or differential equations. As illustrated in Figure 3 for a PEFC-base power plant, steady-state process flowsheets are usually described by lumped parameter models described by algebraic equations. Similarly, dynamic process flowsheets are described by lumped parameter models comprising differential-algebraic equations. Models that deal with spatially distributed models are frequently considered at the device... 

The tubule is a spatially extended structure, and it presents both elastic properties and resistance to the fluid flow. The dynamic pressure and flow variations in such a structure can be represented by a set of coupled partial differential equations [11]. An approximate description in terms of ordinary differential equations (a lumped model) consists of an alternating sequence of elastic and resistive elements, and the simplest possible description, which we will adopt here, applies only a single pair of such elements. Hence our model [12] considers the proximal tubule as an elastic structure with little or no flow resistance. The pressure P, in the proximal tubule changes in response to differences between the in- and outgoing fluid flows ... 

This completes the spatially lumped electrode model, which obviously fits to the input/output interface described earher (Figure 28.3). The electrode model can thus be combined with the balance equations of the MCFC model described in Section 28.2. 

Figure 9. A lumped model to study interactions and spatial patterns in assemblies of catalyst particles [67].

A differential equation for a function that depends on only one variable, often time, is called an ordinary differential equation. The general solution to the differential equation includes many possibilities the boundaiy or initial conditions are needed to specify which of those are desired. If all conditions are at one point, then the problem is an initial valueproblem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinaiy differential equations become two-point boundaiy value problems, which are treated in the next section. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and in all models where there are no spatial gradients in the unknowns. 

We first choose variables sufficient to describe the situation. This choice is tentative, for we may need to omit some or recruit others at a later stage (e.g., if V is constant, it can be dismissed as a variable). In general, variables fall into two groups independent (in our example, time) and dependent (volume and concentration) variables. The term lumped is applied to variables that are uniform throughout the system, as all are in our simple example because we have assumed perfect mixing. If we had wished to model imperfect mixing, we would have had either to introduce a number of different zones (each of which would then be described by lumped variables) or to introduce spatial coordinates, in which case the variables are said to be distributed.2 Lumped variables lead to ordinary equations distributed variables lead to partial differential equations. 

Wang15 investigated heat and mass transport and electrochemical kinetics in the cathode catalyst layer during cold start, and identified the key parameters characterizing cold-start performance. He found that the spatial variation of temperature was small under low current density cold start, and thereby developed the lumped thermal model. A dimensionless parameter, defined as the ratio of the time constant of cell warm-up to that of ice... 

The next level of transient modeling improves the spatial accuracy beyond the lumped analysis given in prior sections by breaking up the original control volume that previously spanned the entire cell into numerous additional control volumes, or nodes (see Figure 9.8). However, forpurposes of computational efficiency, it is noted that once the number of computational nodes becomes greater than 10, additional... 

Mathematical models can also be classified according to the mathematical foundation the model is built on. Thus we have transport phenomena-bas A models (including most of the models presented in this text), empirical models (based on experimental correlations), and population-based models, such as the previously mentioned residence time distribution models. Models can be further classified as steady or unsteady, lumped parameter or distributed parameter (implying no variation or variation with spatial coordinates, respectively), and linear or nonlinear. 

These authors [32, 33] have considered an alternative classification based on the nature of the variables involved in the model. They classify models by grouping them into opposite pairs deterministic vs. probabilistic, linear vs. non-linear, steady vs. non-steady state, lumped vs. distributed parameters models. In a lumped parameters model, variations of some variable (usually a spatial one) are ignored and its value is assumed to be uniform throughout the entire system. On the other hand, distributed parameters models take into account detailed variations of variables throughout the system. In the kinetic description of a chemical system, lumping concerns chemical constituents and has been widely used (see Sects. 2.4 and 2.5). 

Before assessing how a chemical moves in the environment, the relevant media, or compartments, must be defined. The environment can be considered to be composed of four broad compartments—air, water, soil, and biota (including plants and animals)—as shown in Fig. 6.6. Various approaches to modeling the environment have been described.14-16 The primary difference in these approaches is the level of spatial and component detail included in each of the compartments. For example, the most simplistic model considers air as a lumped compartment. A more advanced model considers air as composed of air and aerosols, composed of species such as sodium chloride, nitric and sulfuric acids, soil, and particles released anthropogenically.17 A yet more complex model considers air as composed of air in stratified layers, with different temperatures and accessibility to the earth s surface, and aerosols segmented into different size classes.16 As the model complexity increases, its resolution and the data demands also increase. Andren et al.16 report that the simplest of models with lumped air, water, and soil compartments is suitable for... 

Thus, we recover the Danckwerts model only if no distinction is made between the cup-mixing and spatial average concentrations (with this assumption, the effective axial dispersion coefficient is given by the Taylor-Aris theory). This derivation also shows that the concept of an effective axial dispersion coefficient and lumping the macro- and micromixing effects into one parameter is valid only at steady-state, constant inlet conditions and when the deviation from plug flow is small. [Remark Even with all these constraints, the error in the model because of the assumption (cj) — cym is of the same order of magnitude as the dispersion effect ]... 

The threshold volume fraction, c, depends not only on the spatial dimensionality but also on extrinsic factors. For a 3D, continuous, random mixture, d is ideally 15% values for other dimensions are listed in Table 7. However, c may be easily affected by the state of dispersion, size, shape, orientation, and distribution of the conductive phase. Therefore, it is best determined empirically. Finally, the prefactor o depends on details of ionic and molecular interactions and can only be computed from specific microscopic models. The important feature to be appreciated here is that the topological and geometrical information of the cluster connectivity is lumped together in the (c-c ) factor whereas the microscopic diffusion information and various interactions are contained in the prefactor o. ... 

---