Fundamental model equations

In this section we examine the primary transient phenomena that are of interest to SOFC analysis, and provide the fundamental model equations for each one. Examples for the use of these models are given in later sections. While the focus is on reduced-order models (lumped and one-dimensional), depending on the needs of the fuel cell designer, this may, or may not be justifiable. Each fuel cell model developer needs to ensure that the solution approach taken will provide the information needed for the problem at hand. For the goal of calculating overall cell performance, however, it is often that one-dimensional methods such as outlined below will be viable. 

In addition to the basic continuous column model assumptions of equilibrium stages and adiabatic operation, dynamics-related assumptions are made for the batch model. Distefano (1968) assumed constant volume of liquid holdup, negligible vapor holdup, and negligible fluid dynamic lag. Although different solution strategies may be employed, the fundamental model equations are the same. 

As in all packed-column operations, the fundamental model equations consist of differential balances taken over each phase the principal novelty here is the simultaneous use of mass and energy balances. 

It is interesting to note that the Voigt model is useless to describe a relaxation experiment. In the latter a constant strain was introduced instantaneously. Only an infinite force could deform the viscous component of the Voigt model instantaneously. By constrast, the Maxwell model can be used to describe a creep experiment. Equation (3.56) is the fundamental differential equation of the Maxwell model. Applied to a creep experiment, da/dt = 0 and the equation becomes... 

The random error arises from the measurement of y the true value of which is not known. The measurements are assumed to be free of systematic errors. The modeling equations contain adjustable parameters to account for the fact that the models are phenomenological. For example, kinetic rate expressions contain rate constants (parameters) the value of which is unknown and not possible to be obtained from fundamental principles. 

Model equations. Fundamental process models are very useful in optimizing the design and operation of LPCVD systems. A fundamental model of an LPCVD reactor similar to Figure El4.5a was presented by Jensen and Graves (1983) and included the following simplifying assumptions ... 

The advantage of the MTC model, as opposed to the CRK model, is that the exchange rate constant kp is no longer an empirical constant, but is now defined in terms of more fundamental processes that can be separately modeled. Equations 3.1 and 3.12 are linked via the equality... 

There are two general types of aerosol source apportionment methods dispersion models and receptor models. Receptor models are divided into microscopic methods and chemical methods. Chemical mass balance, principal component factor analysis, target transformation factor analysis, etc. are all based on the same mathematical model and simply represent different approaches to solution of the fundamental receptor model equation. All require conservation of mass, as well as source composition information for qualitative analysis and a mass balance for a quantitative analysis. Each interpretive approach to the receptor model yields unique information useful in establishing the credibility of a study s final results. Source apportionment sutdies using the receptor model should include interpretation of the chemical data set by both multivariate methods. 

In order to make design or operation decisions a process engineer uses a process model. A process model is a set of mathematical equations that allows one to predict the behavior of a chemical process system. Mathematical models can be fundamental, empirical, or (more often) a combination of the two. Fundamental models are based on known physical-chemical relationships, such as the conservation of mass and energy, as well as thermodynamic (phase equilibria, etc.) and transport phenomena and reaction kinetics. An empirical model is often a simple regression of dependent variables as a function of independent variables. In this section, we focus on the development of process models, while Section III focuses on their numerical solution. 

The above derivation assumes that vapor flows upward in plug flow and that there is no horizontal vapor mixing, while liquid flows horizontally in plug flow and there is no vertical mixing. Lockett and Uddin (12,122) and Standart (123,124) showed that these liquid flow assumptions are poor, unnecessary, and lead to incorrect implications regarding tray efficiency. By modifying the definition of NL, Lockett derived a fundamentally superior equation analogous to Eq. (7.13). Most theoretical models, however, use Eq. (7.13). Equation (7.13) is also the equation used for packed columns, but for packed columns, it is based on sounder assumptions (12). 

Since the fundamental rate equation of the diffusion layer model has the typical form of a first-order rate process (5.1), using (5.4) and (5.14), the MDT is found equal to the reciprocal of the rate constant k ... 

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