Mass governing equations

Boundary conditions are special treatments used for internal and external boundaries. For example, the center line in cylindrical geometry is an internal boundary that is modeled as a plane of symmetry. External boundaries model the world outside the mesh. The outermost row of elements is often used to implement the boundary condition as shown in Fig. 9.13. The mass, stress, velocity, etc., of the boundary elements are defined by the boundary conditions rather than the governing equations. External boundary conditions are typically prescribed through user input. 

The governing equations of fluid flow represent matliematieal statements of the eonservation of mass, known as the eontinuity equation ... 

The governing equations for mass flow, energy flow, and contaminant flow in a room will be the continuity equation, Navier-Stokes equations (one in each coordinate direction), the energy equation, and the mass transport equation, respectively. 

Mass transport measurements have shown that cation transport predominates in FeO (Fe ) and Fej04 (Fe, Fe ), whereas anion transport predominates in FejOj (0 ). This leads to the well-accepted growth scheme for multi-layered scale growth on iron shown in Fig. 7.3, with the governing equations for individual layer growth being ... 

In a thin flat platelet, the mass transfer process is symmetrical about the centre-plane, and it is necessary to consider only one half of the particle. Furthermore, again from considerations of symmetry, the concentration gradient, and mass transfer rate, at the centre-plane will be zero. The governing equation for the steady-state process involving a first-order reaction is obtained by substituting De for D in equation 10.172 ... 

We have considered thermodynamic equilibrium in homogeneous systems. When two or more phases exist, it is necessary that the requirements for reaction equilibria (i.e., Equations (7.46)) be satisfied simultaneously with the requirements for phase equilibria (i.e., that the component fugacities be equal in each phase). We leave the treatment of chemical equilibria in multiphase systems to the specialized literature, but note that the method of false transients normally works quite well for multiphase systems. The simulation includes reaction—typically confined to one phase—and mass transfer between the phases. The governing equations are given in Chapter 11. 

There are two levels, discrete particle level and continuum level, for describing and modeling of the macroscopic behaviors of dilute and condensed matters. The physics laws concerning the conservation of mass, momentum, and energy in motion, are common to both levels. For simple dilute gases, the Boltzmann equation, as shown below, provides the governing equation of gas dynamics on the discrete particle level... 

Consider Equations (6-10) that represent the CVD reactor problem. This is a boundary value problem in which the dependent variables are velocities (u,V,W), temperature T, and mass fractions Y. The mathematical software is a stand-alone boundary value solver whose first application was to compute the structure of premixed flames.Subsequently, we have applied it to the simulation of well stirred reactors,and now chemical vapor deposition reactors. The user interface to the mathematical software requires that, given an estimate of the dependent variable vector, the user can return the residuals of the governing equations. That is, for arbitrary values of velocity, temperature, and mass fraction, by how much do the left hand sides of Equations (6-10) differ from zero ... 

In terms of suitably chosen nondimensional variables, the governing equations for a two-dimensional description of the physical system may be written as follows mass balance on the gas phase... 

The modeling of steady-state problems in combustion and heat and mass transfer can often be reduced to the solution of a system of ordinary or partial differential equations. In many of these systems the governing equations are highly nonlinear and one must employ numerical methods to obtain approximate solutions. The solutions of these problems can also depend upon one or more physical/chemical parameters. For example, the parameters may include the strain rate or the equivalence ratio in a counterflow premixed laminar flame (1-2). In some cases the combustion scientist is interested in knowing how the system mil behave if one or more of these parameters is varied. This information can be obtained by applying a first-order sensitivity analysis to the physical system (3). In other cases, the researcher may want to know how the system actually behaves as the parameters are adjusted. As an example, in the counterflow premixed laminar flame problem, a solution could be obtained for a specified value of the strain... 

The governing equation for mass transport in the case of an incompressible flow field is easily derived from the general convection-diffusion equation Eq. (32) with... 

We assume all reactions to be first order and irreversible within the range of the experimental conditions. The governing differential mass balance equations and their solutions have been reported [9J. The values of the constants through at 450°C are shown in Table I. A comparison of the experimental data with the theoretical predictions is shown in Figures 2 through 4 the above assumption of a first order reaction appears reasonable. 

For steady flow of a gas (at a constant mass flow rate) in a uniform pipe, the pressure, temperature, velocity, density, etc. all vary from point to point along the pipe. The governing equations are the conservation of mass (continuity), conservation of energy, and conservation of momentum, all applied to a differential length of the pipe, as follows. 

At this point we can derive a set of governing equations that fully describes the equilibrium state of the geochemical system. To do this we will write the set of independent reactions that can occur among species, minerals, and gases in the system and set forth the mass action equation corresponding to each reaction. Then we will derive a mass balance equation for each chemical component in the system. Substituting the mass action equations into the mass balance equations gives a set... 

The final form of the governing equations is given by substituting the mass action equation (Eqn. 3.27) for each occurrence of ntj in the mass balance equations (Eqns. 3.28-3.31). The substituted equations are,... 

To see how the governing equations might be solved, we consider a system that contains an aqueous fluid and several minerals but has no gas buffer. If we know the system s bulk composition in terms of Mw, Mi, and M, we can evaluate Equations 3.32-3.34 to give values for the unknown variables the solvent mass nw, the basis species molalities m, and the mineral mole numbers nt-... 

Geochemists, however, seem to have reached a consensus (e.g., Karpov and Kaz min, 1972 Morel and Morgan, 1972 Crerar, 1975 Reed, 1982 Wolery, 1983) that Newton-Raphson iteration is the most powerful and reliable approach, especially in systems where mass is distributed over minerals as well as dissolved species. In this chapter, we consider the special difficulties posed by the nonlinear forms of the governing equations and discuss how the Newton-Raphson method can be used in geochemical modeling to solve the equations rapidly and reliably. 

The governing equations are composed of two parts mass balance equations that require mass to be conserved, and mass action equations that prescribe chemical equilibrium among species and minerals. Water Aw, a set of species, 4/, the min-... 

There are, in fact, a number of solutions to the governing equations, but usually (see Chapter 12) only one with positive mole numbers and concentrations. Fortunately, the latter answer is of interest to all but the most abstract-thinking geochemist. The requirement that the iteration produce positive masses is known in chemical modeling as the non-negativity constraint. 

In this chapter we consider how to construct reaction models that are somewhat more sophisticated than those discussed in the previous chapter, including reaction paths over which temperature varies and those in which species activities and gas fugacities are buffered. The latter cases involve the transfer of mass between the equilibrium system and an external buffer. Mass transfer in these cases occurs at rates implicit in solving the governing equations, rather than at rates set explicitly by the modeler. In Chapter 16 we consider the use of kinetic rate laws, a final method for defining mass transfer in reaction models. 

Equation (9.41) constitutes a fundamental solution for purely convective mass burning flux in a stagnant layer. Sorting through the S-Z transformation will allow us to obtain specific stagnant layer solutions for T and Yr However, the introduction of a new variable - the mixture fraction - will allow us to express these profiles in mixture fraction space where they are universal. They only require a spatial and temporal determination of the mixture fraction/. The mixture fraction is defined as the mass fraction of original fuel atoms. It is as if the fuel atoms are all painted red in their evolved state, and as they are transported and chemically recombined, we track their mass relative to the gas phase mixture mass. Since these fuel atoms cannot be destroyed, the governing equation for their mass conservation must be... 

A spray is a turbulent, two-phase, particle-laden jet with droplet collision, coalescence, evaporation (solidification), and dispersion, as well as heat, mass and momentum exchanges between droplets and gas. In spray modeling, the flow of gas phase is simulated typically by solving a series of conservation equations coupled with the equations for spray process. The governing equations for the gas phase include the equations of mass, momentum and energy... 

At the heart of the model are the heat and mass balance equations governing the chlorine gas, brine and amalgam layers within the cell as illustrated by Fig. 20.3. At a more detailed level each cell is divided into eight zones. Conditions within each zone are assumed to be constant and there is a trade-off between model accuracy and execution time associated with the number of zones. Typically eight zones have been found to be a good compromise. 

Consider the sodium nltrate/goethite suspension titrated in the presence and absence of lead ion. The following equations governing the titrations are easily derived from the charge and mass balance equations ... 

The solution procedure to this equation is the same as described for the temporal isothermal species equations described above. In addition, the associated temperature sensitivity equation can be simply obtained by taking the derivative of Eq. (2.87) with respect to each of the input parameters to the model. The governing equations for similar types of homogeneous reaction systems can be developed for constant volume systems, and stirred and plug flow reactors as described in Chapters 3 and 4 and elsewhere [31-37], The solution to homogeneous systems described by Eq. (2.81) and Eq. (2.87) are often used to study reaction mechanisms in the absence of mass diffusion. These equations (or very similar ones) can approximate the chemical kinetics in flow reactor and shock tube experiments, which are frequently used for developing hydrocarbon combustion reaction mechanisms. 

---