The Fundamental Model Equations

The momentum balance can be obtained by application of Newton s second law on a moving fluid element. Over a volume element of a chemical reactor, the balance of momentum in direction / can be written as  

In chemical reactors, only pressure drop and friction forces have to be considered in most cases. A number of specific pressure drop equations are discussed in the chapters on tubular plug flow reactors and on fixed bed catalytic reactors. 

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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... 

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 ... 

Development of the Nonlinear Fluorescence Quenching Model. The main function of the nonlinear model is to relate aquatic equilibrium considerations based on conventional solution thermodynamics to observed fluorescence intensity changes occurring as metal ion is added at fixed pH. The fundamental relationship equating fluorescence quenching to complexation is present in equation 1. 

In industrial reactors there are normally gradients in the species mass concentrations, temperature, pressure and velocity in all space directions. The fundamental microscopic equations give a detailed description of all the known mechanisms involved. In the chemical reactor engineering approach we desire to eliminate the mechanisms that is not essential for the reactor performance from the equations to reduce the computational demand. An appropriate engineering packed bed reactor model is thus tailored for its main purpose. It is as simple as possible, but still include a sufficient representation of essential mechanisms involved. 

Another structure/function transport model, often referred to as the capillary or electroki-netic model, predefines the microlevel structure of an ion-exchange membrane as an array of pores of known dimensions with a specified distribution of ion-exchange sites on the pore walls. Equations describing solute and solvent transport and theories for molecularlevel ion/solvent and ion-membrane interactions are then generated, based on this pore structure [151], The fundamental transport equation for the molar flux of ionic species is the Nemst-Planck equation... 

The wedge effect is significant for high energetic radiation in low-Z matrices. The effect can be modelled adequately by taking it into account in the Fundamental Parameter equations. 

Although the fundamental mathematical equations describing a physical phenomenon are often very compact, as for example the Schrodinger equation written in operator form HT = E , their application to all but the simplest model systems usually leads to equations that cannot be solved in analytical form. Even if the equations could be solved, one may only be interested in the solution for a certain limited range of variables. In many cases, it is therefore of interest to obtain an approximate solution, and preferably in a form where the accuracy of the solution can be improved in a systematic fashion. We will here consider three approaches for obtaining such approximate solutions ... 

Instead of applying the fundamental conservation equations, as described above, another modeling approach is to characterize gas turbine performance by utilizing real steady state engine performance data, as in (Hung, 1991). It is assumed that transient thermodynamic and flow processes are characterized by a continuous progression along the steady state performance curves. This is known as the quasi-static assumption. The dynamics of the gas turbine, e.g. combustion delay, motor inertia, fuel pump lag etc. are then represented as lumped quantities separate from the steady-state performance curves. Very simple models result if it is further assumed that the gas turbine is operated at all times close to rated speed (Rowen, 1983). 

Pick s second law of diffusion is for a non-steady state or transient conditions in which dCj/t 0. Using Crank s model [23] for the rectangular element shown in Figure 4.1 yields the fundamental differential equations for the rate of concentration. Consider the central plane as the reference point in the rectangular volume element and assume that the diffusing plane at position 2 moves along the x-direction at a distance x-da from position 1 and x-Hdi to position 3. Thus, the rate of diffusion that enters the volume element at position 1 and leaves at position 3 is... 

Attention is especially drawn to the fact that any procedure or procedures that modify or massage the experimental data so that they conform to the stringent conditions of Equation (7) and hence to Equation (1) do not cure the underlying inadequacies in the fundamental model postulates that led to Equation (1). Such procedures only serve to obscure the inadequacies of the theoretical model and are used to enhtmce the belief that the theory is correct and experimentally proven. ... 

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