Conservation equations, defined

The two sets of conservative equations defined by Eq. (8.2) and Eqs. (8.2)-(8.2) describe turbulent reacting two-phase flows that require a high grid resolution in order to solve from the smallest to the largest scales. In LES only the largest scales are computed while the smallest scales are modeled. 

Peclet Number, Pe dimensionless number appearing in enthalpy or species mass conservation equations (defined for heat transfer and mass transfer, respectively). It is interpreted again as the ratio of the convective transport to the molecular transport and is defined as... 

These expressions are inserted in the conservation equations, and the boundary conditions provide a set of relationships defining the U and V coefficients [125-129]. 

Bidispersed Particles For particles of radius Cp comprising adsorptive subparticles of radius r, that define a macropore network, conservation equations are needed to describe transport both within the macropores and within the subparticles and are given in Table 16-11, item D. Detailed equations and solutions for a hnear isotherm are given in Ruthven (gen. refs., p. 183) and Ruckenstein et al. [Chem. Eng. Sci., 26, 1306 (1971)]. The solution for a linear isotherm with no external resistance and an infinite fluid volume is ... 

The properties required of a material in order for it to support a stable shock wave were listed and discussed. Rarefaction, or release waves were defined and their behavior was described. The useful tool of plotting shocks, rarefactions, and boundaries in the time-distance plane (the x-t diagram) was introduced. The Lagrangian coordinate system was defined and contrasted to the more familiar Eulerian coordinate system. The Lagrangian system was then used to derive conservation equations for continuous flow in one dimension. 

The Navier-Stokes equation defines a set of three relations for four unknown quantities, iq, Uj, M3 and p. Another equation is needed to close the set, which is the equation of mass conservation ... 

The situation is different for incompressible flow. In that case, no equation of motion for the pressure field exists and via the mass conservation equation Eq. (17) a dynamic constraint on the velocity field is defined. The pressure field entering the incompressible Navier-Stokes equation can be regarded as a parameter field to be adjusted such that the divergence of the velocity field vanishes. 

The mathematical model describing the two-phase dynamic system consists of modeling of the flow and description of its boundary conditions. The description of the flow is based on the conservation equations as well as constitutive laws. The latter define the properties of the system with a certain degree of idealization, simplification, or empiricism, such as equation of state, steam table, friction, and heat transfer correlations (see Sec. 3.4). A typical set of six conservation equations is discussed by Boure (1975), together with the number and nature of the necessary constitutive laws. With only a few general assumptions, these equations can be written, for a one-dimensional (z) flow of constant cross section, without injection or suction at the wall, as follows. 

For a given mass transfer problem, the above conservation equations must be complemented with the applicable initial and boundary conditions. The problem of finding the mathematical function that represents the behaviour of the system (defined by the conservation equations and the appropriate set of initial and boundary conditions), is known as a boundary value problem . The boundary conditions specifically depend on the nature of the physicochemical processes in which the considered component is involved. Various classes of boundary conditions, resulting from various types of interfacial processes, will appear in the remainder of this chapter and Chapters 4 and 10. Here, we will discuss some simple boundary conditions using examples of the diffusion of a certain species taken up by an organism ... 

P is positive since precipitation decreases the amount of species i in the matrix. PJ is defined in a similar way. Two elements i and j give the system of conservation equations... 

The first equation defines the ionization constant of water at 25 °C (we omit the sign of the charges to simplify notation). The second is the same as Eq. (2.6.3), while the third is the conservation of the total initial concentration of the (weak) acid Nj- (we assume that there is no change in volume during the titration, hence this is the same as the conservation of the total number of acid molecules). The fourth equation is the electroneutrality condition, where [iV ] is the concentration of the added (strong) base. 

As noted in the Introduction, one of the defining characteristics of any fuel-cell model is how it treats transport. Thus, these equations vary depending on the model and are discussed in the appropriate subsections below. Similarly, the auxiliary equations and equilibrium relationships depend on the modeling approach and equations and are introduced and discussed where appropriate. The reactions for a fuel cell are well-known and were introduced in section 3.2.2. Of course, models modify the reaction expressions by including such effects as mass transfer and porous electrodes, as discussed later. Finally, unlike the other equations, the conservation equations are uniformly valid for all models. These equations are summarized below and not really discussed further. 

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 ... 

All conservation equations in continuum mechanics can be derived from the general transport theorem. Define a variable F(t) as a volume integral over an arbitrary volume v(t) in an r-space... 

The conservation equations for mass and enthalpy for this special situation have already been given with eqs 76 and 62. As there is no diffusional mass transport inside the pellet, the overall catalyst effectiveness factor is identical to the film effectiveness factor i/cxl which is defined as the ratio of the effective reaction rate under surface conditions divided by the intrinsic chemical rate under bulk fluid phase conditions (see eq 61). For an nth order, irreversible reaction we have the following expression ... 

A differential characteristic which demands a lower degree of standardization is the reaction rate. The rate of a chemical reaction with respect to compound B at a given point is defined as the rate of formation of B in moles per unit time per unit volume. It cannot be measured directly and is determined from the rates of change of some observable quantities such as the amount of substance, concentration, partial pressure, which are subject to measurements. Reaction rates are obtained from observable quantities by use of the conservation equations resulting from the mass balance for the given reactor type. 

When Boussinesq approximation is adopted in full conservation equations, it is noted that the effect of buoyancy force appears in terms of GrjRe where Gr is the Grashof number and Re is the Reynolds number defined in terms of appropriate length, velocity and temperature scales. However, Leal et al. (1973) and Sparrow Minkowycz (1962) have shown that the equivalent buoyancy parameter with the boundary layer assump-... 

Equation (33) is a dimensionless form of the species-conservation equation. We are entitled to indicate that F (t, (p as defined in equation (34), is a function of t and q>, because the additional variables p and F, which appear on the right-hand side of equation (34), are easily expressed in terms of t, (p, and constants through equations (6) and (27). The function F (t, (p) is nonnegative over the entire range 0 < t < 1, in the physically acceptable range of (p, and it equals zero only at t = 1, although it becomes very small (because of the exponential factor involving T) near t = 0 where T is small. 

By definition, P(v) possesses the necessary nonnegative and normalization properties of probability-density functions. It is especially useful in connection with the Favre-averaged formulation of the conservation equations, since corresponding averages are obtained from P(v) by the usual rules for averaging. Thus, for the v and v" defined above equation (2),... 

Derivations of conservation equations from the viewpoint of kinetic theory usually do not exhibit explicitly the diffusion terms, such as diffusion stresses, that appear on the right-hand sides of equations (49), (50), and (51), since it is unnecessary to introduce quantities such as afj specifically in these derivations. Kinetic-theory developments work directly with the left-hand sides of equations (49), (50), and (51). Transport coefficients (Appendix E) are defined only in terms of these kinetic-theory quantities because prescriptions for calculating the individual continua transports, afj and qf, are unduly complex. Moreover, measurement of diffusion stresses is feasible only by direct measurement of diffusion velocities, followed by use of equation (24). Therefore, it has not been fruitful to study the diffusion terms which, in a sense, may be viewed as artifacts of the continuum approach. 

In flow problems, the conservation equations often can be written in forms involving dimensionless ratios of various transport coefficients. These ratios are defined in the present section. They refer to molecular transport, not to corresponding quantities that often have been defined for turbulent flows. 

In many of the arguments involving chemostats it was shown that the omega limit set had to lie in a restricted set, and the equations were analyzed on that set one simply could choose initial conditions in the restricted set at time zero. The equation defining the restricted set - in effect, a conservation principle - allowed one variable to be eliminated from the system. We want to abstract this idea and make it rigorous. The omega limit set lies in a lower-dimensional set, and the trajectories in that set satisfy a smaller system of differential equations. However, it is not clear (and, indeed, not true [T3]) that the asymptotic behavior of the two systems is necessarily the same. (A very nice paper of Thieme [Tl] gives examples and helpful theorems for asymptotically autonomous systems. A classical result in this direction is a paper of Markus [M].) In this appendix, a theorem is presented which justifies the procedure on the basis of stability. 

The reference level is defined by the composition of a pure solution of HA in H2O (/ = 0 [ANC] = 0), which is defined by the proton condition, [H l = [A-] -i- [OH ]. (In this and subsequent equations, the charge type of the eicid is unimportant the equation defining the net proton excess or deficiency can always be derived from a combination of the concentration condition and the condition of electroneutrality.) Thus in a solution containing a mixture of HA and NaA, [ANC] is a conservative capacity parameter. It must be expressed in concentrations (and not activities). Addition of HA (a species defining the reference level) does not change the proton deficiency and thus does not affect [ANC]. 

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