Species continuity equations

Mechanism. The thermal cracking of hydrocarbons proceeds via a free-radical mechanism (20). Siace that discovery, many reaction schemes have been proposed for various hydrocarbon feeds (21—24). Siace radicals are neutral species with a short life, their concentrations under reaction conditions are extremely small. Therefore, the iategration of continuity equations involving radical and molecular species requires special iategration algorithms (25). An approximate method known as pseudo steady-state approximation has been used ia chemical kinetics for many years (26,27). The errors associated with various approximations ia predicting the product distribution have been given (28). 

With the introduction of Gear s algorithm (25) for integration of stiff differential equations, the complete set of continuity equations describing the evolution of radical and molecular species can be solved even with a personal computer. Many models incorporating radical reactions have been pubHshed. 

It can be noted that other approaches, based on irreversible continuum mechanics, have also been used to study diffusion in polymers [61,224]. This work involves development of the species momentum and continuity equations for the polymer matrix as well as for the solvent and solute of interest. The major difficulty with this approach lies in the determination of the proper constitutive equations for the mixture. Electric-field-induced transport has not been considered within this context. 

The objective of most of the theories of transport in porous media is to derive analytical or numerical functions for the effective diffusion coefficient to use in the preceed-ing averaged species continuity equations based on the structure of the media and, more recently, the structure of the solute. 

In order to illustrate the effects of media structure on diffusive transport, several simple cases will be given here. These cases are also of interest for comparison to the more complex theories developed more recently and will help in illustrating the effects of media on electrophoresis. Consider the media shown in Figure 18, where a two-phase system contains uniform pores imbedded in a matrix of nonporous material. Solution of the one-dimensional point species continuity equation for transport in the pore, i.e., a phase, for the case where the external boundaries are at fixed concentration, Ci and Cn, gives an expression for total average flux... 

Michaels [241], using a model pore shown in Figure 19, considered the constriction effect by solving the steady-state species continuity equations in a model pore consisting of a single constriction. The result for the pore shown in Figure 19 is given by... 

Analytical solutions for the closure problem in particular unit cells made of two concentric circles have been developed by Chang [68,69] and extended by Hadden et al. [145], In order to use the solution of the potential equation in the determination of the effective transport parameters for the species continuity equation, the deviations of the potential in the unit cell, defined by... 

Determination of the effective transport coefficients, i.e., dispersion coefficient and electrophoretic mobility, as functions of the geometry of the unit cell requires an analogous averaging of the species continuity equation. Locke [215] showed that for this case the closure problem is given by the following local problems ... 

It is important to note that the closure problem for the species continuity equation requires solutions for the deviations of the potential, i.e the /and g fields, anfi knowledge of the average potential ( ). This result is very similar to that found by the area averaging method in Sauer et al. [345], Utilizing the closure expressions the average species continuity equation becomes... 

The steady-state continuity equations which describe mass balance over a fluid volume element for the species in the stagnant film which are subject to uniaxial diffusion and reaction in the z direction are... 

For modeling a continuous pelletizer, it is advantageous to formulate the snowballing kinetics in the well-known continuity equation for the pellet species... 

These convective transport equations for heat and species have a similar structure as the NS equations and therefore can easily be solved by the same solver simultaneously with the velocity field. As a matter of fact, they are much simpler to solve than the NS equations since they are linear and do not involve the solution of a pressure term via the continuity equation. In addition, the usual assumption is that spatial or temporal variations in species concentration and temperature do not affect the turbulent-flow field (another example of oneway coupling). 

A stoichiometric analysis based on the species expected to be present as reactants and products to determine, among other things, the maximum number of independent material balance (continuity) equations and kinetics rate laws required, and the means to take into account change of density, if appropriate. (A stoichiometric table or spreadsheet may be a useful aid to relate chosen process variables (Fj,ch etc.) to a minimum set of variables as determined by stoichiometry.)... 

A material balance analysis taking into account inputs and outputs by flow and reaction, and accumulation, as appropriate. This results in a proper number of continuity equations expressing, fa- example, molar flow rates of species in terms of process parameters (volumetric flow rate, rate constants, volume, initial concentrations, etc.). These are differential equations or algebraic equations. 

The solution of this set of equations, 18.4-26 (with expression (A) incorporated) to -29, must be coupled with the set of three independent material-balance or continuity equations to determine the concentration profiles of three independent species, and the temperature profile, for either a specified size (V) of reactor or a specified amount of reaction. A nu-merical solution of the coupled differential equations and property relations is required. Equations (A), (B), and (C) in Example 18-6 illustrate forms of the continuity equation. 

For a complex system, the continuity equation 21.6-1 must be written for more than one species, and equations 21.6-4 and -5 are not sufficient by themselves. Thus, instead of 21.6-4, it is more appropriate to develop profiles for the proper number of concentrations (q) or molar flow rates (F ) see problem 21-21. However, we restrict attention to simple systems in this section. 

We illustrate the development of the model equations for a network of two parallel reactions, A -> B, and A - C, with kt and representing the rate constants for the first and second reactions, respectively. Continuity equations must be written for two of the three species. Furthermore, exchange coefficients (Kbc and Kce) must be determined for each species chosen (here, A and B). 

The high degree of packing of the organisms within a volume can lead to the formation of floes (suspended aggregates), where millions of cells cluster to form particles with dimensions in the order of millimetres [29]. Models for uptake by such ecosystems also assume sphericity, and start from a continuity equation accounting for the consumption of the species throughout the floe ... 

There is a nonzero mass source in the continuity equation, Sm, arising from the summation of all species equations. A general form of this source term is given in Table 1. Use has been made of the assumption that summation of interspecies diffusion within the gas phase is equal to zero. Specifically, one has... 

B. Heat and Mass Transfer 1. The Species Continuity Equation... 

Equation (1-47) is identical in form to the species continuity equation, Eq. (1 -38), and this leads to close analogies between heat and mass transfer as discussed in the next section. 

Parallel to the boundary conditions discussed above for the species continuity equation, we consider in this book only uniform temperature on the surface of the particle, uniform temperature in the continuous phase remote from the particle and uniform initial temperatures in each phase. Hence... 

Very few solutions have been obtained for heat or mass transfer to nonspherical solid particles in low Reynolds number flow. For Re = 0 the species continuity equation has been solved for a number of axisymmetric shapes, while for creeping flow only spheroids have been studied. 

For constant-property steady flow the species continuity equation, Eq. (1-38), becomes... 

Similarly, the continuity equation for species i is readily shown (12) to reduce to the form... 

In order to calculate the density of reactant B about A, it is necessary to know by what means the reactants migrate in solution. Under most circumstances, diffusion is a very adequate description (the limitations of and complications to diffusion are discussed in Sect. 6, Chap. 8 Sect. 2 and Chap. 11). In this simple analysis of diffusion, Fick s laws will be used with little further justification, save to note that Fick s second law is identical to the equation satisfied by a random walk function. Hardly a surprising result, because diffusion is a random walk with no retention of information about where the diffusing species was before its current location. In Chap. 3 Sect. 1, the diffusion equation is derived from thermodynamic considerations and the continuity equation (law of conservation of mass). 

As long as the Mach number is small—meaning the velocities are small compared to the sound speed—it is reasonable to assume that the incompressible continuity equation is a good approximation for isothermal, single-species flow. That is, velocity variations have little effect on density variations. As a result the simplifications associated with V-V 0 can be enjoyed. In practical terms, most consider that flows with Ma < 0.3 can be assumed to be gas-dynamically incompressible. 

To derive the species-continuity equations that follow, it is important to establish some relationships between mass fluxes and species concentration fields. At this point the needed relationships are simply stated in summary form. The details are discussed later in chapters on thermochemical and transport properties. 

The evaluation of transport properties, including diffusion coefficients, is the subject of Chapter 12. The objective in this section is only to provide a brief discussion to assist understanding of the following derivation of the species continuity equations. 

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