The Species Continuity Equations

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... 

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... 

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. 

In this form the species-continuity equation is written as... 

Using Eq. 2.30, we could have written the species-continuity equation as... 

This equation shows that the rate of change of species in the mixture contributes directly to the enthalpy change. Recall from the species continuity equation, Eq. 3.124, that there are two principal contributions to the rate of change of chemical species molecular diffusion across the control surfaces and homogeneous chemical reaction within the control volume. Substituting the species continuity equation, Eq. 3.124, yields... 

It is important to determine the partial-differential-equation order. One of the most important reasons to understand order relates to consistent boundary-condition assignment. All the equations are first order in time. The spatial behavior can be a bit trickier. The continuity equation is first order in the velocity and density. The momentum equations are second order on the velocity and first order in the pressure. The species continuity equations are essentially second order in the composition (mass fraction Yy), since (see Eq. 3.128)... 

The objective is to derive a system of equations in general vector form that describes the overall gas-phase mass continuity and the species continuity equations for A and all other species k in the mixture. Assume that there is convective and diffusive transport of the species, but no chemical reaction. 

Use the overall mass-continuity equation to rewrite the species continuity equations, introducing the substantial-derivative operator. Discuss the differences between the two forms of the species-continuity equations. 

Derive the species continuity equation for the CO mass fraction, which should be... 

In the spirit of the Graetz problem (i.e., impose a parabolic velocity profile) develop a nondimensional form of the species-continuity equation. Use the following scale factors and dimensionless variables ... 

Develop and discuss a numerical method to solve the species continuity equation. Write a simulation program that can be used to explore the effects of parameter variations on the behavior of the flow. 

With some significant simplifying assumptions, the species-continuity equation can be put into a form that is analogous to the thermal-energy equation. Specifically, consider that there is no gas-phase chemistry and that a single species, A, is dilute in an inert carrier gas, B. In this case, considering Eq. 3.128, Eq. 6.24 reduces to... 

The species-continuity equation for Y presumes a single species that is dilute in a carrier gas. 

In all of these situations, homogeneous reactions in the gas phase provide source and sink terms in the species continuity equation. In addition the creation and destruction of species can be an important heat source or sink term in the energy equation. Therefore it is important to understand the factors that govern gas-phase chemical kinetics. 

Note that the species continuity equation has been substituted for dYk/dt. 

Substituting these definitions and the species continuity equation, the energy equation for the constant-pressure system becomes... 

Next, we have to solve for the Yj s from the species continuity equations, Equation (32). Unfortunately, these equations cannot be integrated by a similar simple point iteration scheme as they are mathematically "stiff"16 and iterative approaches are unstable. To solve these simultaneous equations, we turn to a perturbation analysis developed by Newman17 where the equations are linearized about an initial guess, and the resulting linear equations are solved numerically. The solution is then used as the next guess, and the linear equations are resolved. The procedure is repeated until the solution no longer changes. 

We see, as for the species continuity equations, that when we perform the averaging procedure we obtain new dependent variables, in this case the terms pu iu j. These new terms are usually called the turbulent momentum fluxes or the Reynolds stresses, but, as before, we have more dependent variables than equations. Thus, some means to evaluate the turbulent momentum fluxes must be developed. Although no rigorous method of obtaining a closed set of equations is known, a number of semi-empirical approaches have been proposed which yield qualitative or semi-quantitative results for appropriately chosen classes of prr blems. 

Appendix B provides the derivation of the design equation from the species continuity equation. In Section 4.1, we carry out macroscopic species balances to derive the species-based design equation of any chemical reactor. In Section 4.2,... 

We derived (in Appendix B) the species continuity equations, describing the variations in species concentrations at any point in the reactor. 

More comprehensive treatments of the species continuity equations and the molar... 

Applications of the species continuity equation in reacting systems is found in ... 

To describe the operation of a chemical reactor, we integrate the species continuity equation over the reactor volume. Multiplying each term in Eq. B.3 by dV and integrating,... 

We consider a two-phase system consisting of a fluid phase and a solid phase as illustrated in Figure 1.7. Here we have identified the fluid phase as the y-phase and the solid phase as the K-phase. The foundations for the analysis of diffusion and reaction in this two-phase system consist of the species continuity equation in the y-phase and the species jump condition at the catalytic surface. The species continuity equation can be expressed as... 

Problems of isothermal mass transfer and reaction are best represented in terms of the species continuity equation and the associated jump condition. We repeat these two... 

Even though equation 1.69 is considered to be the preferred form of the species continuity equation, it is best to begin the averaging procedure with equation 1.35, and we express the superficial average of that form as... 

If the gas phase (e.g., water vapor and air) is considered to be ideal, the overall continuity equation in molar units r uces to a statement that the total molar flux Na + Mb) >s independent of z. Since there is symmetry about the centerline of the slab, Ma -F Mb is zero at z = 0 and therefore is zero everywhere. The species continuity equation in this case reduces to... 

Equations (7.3.1.1-1) and (7.3.1.1-5) are, in fact, extensions of the species continuity equations used in previous chapters, where the flow terms were normally not present. These somewhat detailed derivations have been used to carefully illustrate the development of the equations of transport processes into forms needed to describe chemical reactors, it is seldom that the full equations have to be utilized, and normally only the most important terms will be retained in practical situations. However, (7.2-1) or (7.3.1.1-5) are useful to have available as a fundamental basis. 

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