Simplified conservation equations

In the remainder of the chapter, attention will be restricted to steady-state, one-dimensional, consiant-area flows in which all particles travel at the same velocity and have the same chemical composition. Hence, djdt = 0, Vjj d/dx, d/dv, M = 1 (and the subscript j will usually be omitted), and 

A few additional minor assumptions which are usually valid will appear in the following development. 

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Accordingly, a good starting point for the description of most chemical reactor systems is the following set of simplified conservation equations written in dimensionalized form ... 

Lagranglan codes are characterized by moving the mesh with the material motion, u = y, in (9.1)-(9.4), [24]. The convection terms drop out of (9.1)-(9.4) simplifying all the equations. The convection terms are the first terms on the right-hand side of the conservation equations that give rise to fluxes between the elements. Equations (9.1)-(9.2) are satisfied automatically, since the computational mesh moves with the material and, hence, no volume or mass flux occurs across element boundaries. Momentum and energy still flow through the mesh and, therefore, (9.3)-(9.4) must be solved. 

The Navier-Stokes equation [Eq. (1)] provides a framework for the description of both liquid and gas flows. Unlike gases, liquids are incompressible to a good approximation. For incompressible flow, i.e. a constant density p, the Navier-Stokes equation and the corresponding mass conservation equation simplify to... 

Using the same assumptions that were made in the vapor-layer model, the energy-conservation equation for the incompressible 2-D vapor phase can be simplified to a 1-D equation in boundary layer coordinates ... 

Later we shall include combustion and flame radiation effects, but we will still maintain all of assumptions 2 to 5 above. The top-hat profile and Boussinesq assumptions serve only to simplify our mathematics, while retaining the basic physics of the problem. However, since the theory can only be taken so far before experimental data must be relied on for its missing pieces, the degree of these simplifications should not reduce the generality of the results. We shall use the following conservation equations in control volume form for a fixed CV and for steady state conditions ... 

Simplified mathematical models These models typically begin with the basic conservation equations of the first principle models but make simplifying assumptions (typically related to similarity theory) to reduce the problem to the solution of (simultaneous) ordinary differential equations. In the verification process, such models must also address the relevant physical phenomenon as well as be validated for the application being considered. Such models are typically easily solved on a computer with typically less user interaction than required for the solution of PDEs. Simplified mathematical models may also be used as screening tools to identify the most important release scenarios however, other modeling approaches should be considered only if they address and have been validated for the important aspects of the scenario under consideration. 

A simplified procedure for design is to assume that both tj and — AH/Cp are constant. If, then, eqn. (60) (the heat conservation equation) is divided by eqn. (59) (the mass conservation equation) and integrated, one immediately obtains... 

The conservation equation for receptors defines the total number of receptors as the sum of bound and free receptors (Equation (6.3)). Although the receptor population is in fact made up of subpopulations of receptors in high- and low-affinity states, this is most relevant for modeling agonist interactions. Because most tracers are radiolabeled antagonists, this simplified model is sufficient for most tracer studies. The conservation and mass action equations (Equations (6.3) and (6.4)) can be rearranged to calculate the number of bound receptors ... 

Let us consider a simplified flow, that is, a one-dimensional steady-state flow-without viscous stress or a gravitational force. The conservation equations of continuity, momentum, and energy are represented by rate of mass in - rate of mass out = 0... 

At steady-state operation ( dC/t = dT/dt = 0), the conservation equations presented above simplify to... 

The crystalline inorganic monopropellants decompose directly from the solid to the vapor phase and are approximately described by the above mentioned theoretical work, in spite of the fact that the gas phase processes are simplified. However, the double-base propellants and other organic materials liquefy before vaporizing. In their combustion, so-called foam and fizz zones occur before the vapor phase processes. Much work has been done attempting to apply the conservation equations to the series of processes. This work forms the basis for the summary by Geckler (G3). It is the viewpoint of this author that too many parameters are determined empirically in this application of the theory, so that useful extrapolations are not possible. One must admire the manipulative skill of the early workers in this field and also their determination to formulate a complete theory. When and if the rate parameters become available, a useful theory will be developed with the aid of this early work. 

By neglecting body forces applied to the fluid, momentum conservation (Equation (3.3)) is simplified to ... 

One-dimensional Flow Many flows of great practical importance, such as those in pipes and channels, are treated as onedimensional flows. There is a single direction called the flow direction velocity components perpendicular to this direction are either zero or considered unimportant. Variations of quantities such as velocity, pressure, density, and temperature are considered only in the flow direction. The fundamental conservation equations of fluid mechanics are greatly simplified for one-dimensional flows. A broader category of one-dimensional flow is one where there is only one nonzero velocity component, which depends on only one coordinate direction, and this coordinate direction may or may not be the same as the flow direction. 

As in the classical Poisseulle flow, the y component of velocity will be zero, so that the overall mass continuity equation is identically satisfied. For a steady-state flow, we can write the simplified governing equations describing the velocity, temperature, and species conservation fields. 

Let us indicate that a supplementary flux must be added to take into account the neural flux which is re-injected at the discontinuity v = c (reset potential) and u = u + d after it flowed through the value v = 30 mV (maximum of the action potential) which adds the term J(s,u — d, t) on the right side of the conservation equation. This equation is complex and for the sake of progressive validation, we first consider an uncoupled neural population which means that the interaction flux is null and simplifies the conservation equation ... 

Comparison of the integration of this last conservation equation with the direct simulation of an uncoupled neural population with a given distribution of states (v, u) at the initial time provides a validation for the simplified version of the model. Then adding the imposed flux accounting for connectivity allows us to simulate a large population of Izhikevich neurons with a given pattern of connectivity (number of afferents per neuron and delays kernel). 

Of a more complete approach are the zone models [3], which consider two (or more) distinct horizontal layers filling the compartment, each of which is assumed to be spatially uniform in temperature, pressure, and species concentrations, as determined by simplified transient conservation equations for mass, species, and energy. The hot gases tend to form an upper layer and the ambient air stays in the lower layers. A fire in the enclosure is treated as a pump of mass and energy from the lower layer to the upper layer. As energy and mass are pumped into the upper layer, its volume increases, causing the interface between the layers to move toward the floor. Mass transfer between the compartments can also occur by means of vents such as doorways and windows. Heat transfer in the model occurs due to conduction to the various surfaces in the room. In addition, heat transfer can be included by radiative exchange between the upper and lower layers, and between the layers and the surfaces of the room. 

Conservation of momentum may be used to simplify the equation of motion for the Brownian gas for long time behavior t /I At this regime, the Brownian gas will reach an internal equilibrium with the heat bath. From Eq. (7.188) and the mean velocity in Eq. (7.190), the equation of motion for the mean velocity becomes... 

The differential forms of the conservation equations derived in the appendixes for reacting mixtures of ideal gases are summarized in Section 1.1. From the macroscopic viewpoint (Appendix C), the governing equations (excluding the equation of state and the caloric equation of state) are not restricted to ideal gases. Most of the topics considered in this book involve the solutions of these equations for special flows. The forms that the equations assume for (steady-state and unsteady) one-dimensional flows in orthogonal, curvilinear coordinate systems are derived in Section 1.2, where specializations accurate for a number of combustion problems are developed. Simplified forms of the conservation equations applicable to steady-state problems in three dimensions are discussed in Section 1.3. The specialized equations given in this chapter describe the flow for most of the combustion processes that have been analyzed satisfactorily. 

There are a number of possible approaches to the calculation of influences of finite-rate chemistry on diffusion flames. Known rates of elementary reaction steps may be employed in the full set of conservation equations, with solutions sought by numerical integration (for example, [171]). Complexities of diffusion-flame problems cause this approach to be difficult to pursue and motivate searches for simplifications of the chemical kinetics [172]. Numerical integrations that have been performed mainly employ one-step (first in [107]) or two-step [173] approximations to the kinetics. Appropriate one-step approximations are realistic for limited purposes over restricted ranges of conditions. However, there are important aspects of flame structure (for example, soot-concentration profiles) that cannot be described by one-step, overall, kinetic schemes, and one of the major currently outstanding diffusion-flame problems is to develop better simplified kinetic models for hydrocarbon diffusion flames that are capable of predicting results such as observed correlations [172] for concentration profiles of nonequilibrium species. 

The reader will note that all the conservation equations except the species equation have been integrated once as a consequence of our simplifying assumptions. The final equations of the present section are used in the following applications. 

The presentation of the subject of spray combustion in Chapter 11 is not greatly different from that in the first edition. An updated outlook on the subject has been provided, and the formulation has been generalized to admit time dependences in the conservation equations. The analysis of spray deflagration has been abbreviated, and qualitative aspects of the results therefrom have been anticipated on the basis of simplified physical reasoning. In addition, brief discussions of the topics of spray penetration and of cloud combustion have been added. 

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