Continuity equation overall

Material Balances Whenever mass-transfer applications involve equipment of specific dimensions, flux equations alone are inadequate to assess results. A material balance or continuity equation must also be used. When the geometiy is simple, macroscopic balances suffice. The following equation is an overall mass balance for such a unit having bulk-flow ports and ports or interfaces through which diffusive flux can occur ... 

Trinh et al. [399] derived a number of similar expressions for mobility and diffusion coefficients in a similar unit cell. The cases considered by Trinh et al. were (1) electrophoretic transport with the same uniform electric field in the large pore and in the constriction, (2) hindered electrophoretic transport in the pore with uniform electric fields, (3) hydrodynamic flow in the pore, where the velocity in the second pore was related to the velocity in the first pore by the overall mass continuity equation, and (4) hindered hydrodynamic flow. All of these four cases were investigated with two different boundary condi-... 

In the simplest incompressible flow problems under constant property conditions, the velocity and pressure fields (u and p) are the unknowns. In principle, Eq. (1-1) and the overall continuity equation, Eq. (1-9) below, are sufficient for... 

In the thin layer adjacent to the particle surface the overall continuity equation may be written (SI)... 

We also account for density, heat capacity, and molecular weight variations due to temperature, pressure, and mole changes, along with temperature-induced variations in equilibrium constants, reaction rate constants, and heats of reaction. Axial variations of the fluid velocity arising from axial temperature changes and the change in the number of moles due to the reaction are accounted for by using the overall mass conservation or continuity equation. 

The continuity equation is a statement of mass conservation. As presented in Section 3.1, however, no distinction is made as to the chemical identity of individual species in the flow. Mass of any sort flowing into or out of a differential element contributes to the net rate of change of mass in the element. Thus the overall continuity equation does not need to explicitly demonstrate the fact that the flow may be composed of different chemical constituents. Of course, the equation of state that relates the mass density to other state variables does indirectly bring the chemical composition of the flow into the continuity equation. Also, as presented, the continuity-equation derivation does not include diffusive flux of mass across the differential element s surfaces. Moreover there is no provision for mass to be created or destroyed within the differential element s volume. 

When considering the mass continuity of an individual species in a multicomponent mixture, there can be, and typically is, diffusive transport across the control surfaces and the production or destruction of an individual species by volumetric chemical reaction. Despite the fact that individual species may be transported diffusively across a surface, there can be no net mass that is transported across a surface by diffusion alone. Moreover homogeneous chemical reaction cannot alter the net mass in a control volume. For these reasons the overall mass continuity need not consider the individual species. At the conclusion of this section it is shown that that the overall mass continuity equation can be derived by a summation of all the individual species continuity equations. 

As discussed earlier in Eqs. 3.93 and 3.121 both terms on the right-hand side are zero. Also, since f=1 Yk = 1, the overall mass-continuity equation is recovered,... 

It may also be noted that summing the system representation must also produce the starting point for derivation of the overall mass-continuity equation,... 

It is interesting to note that if the starting point had been Eq. 3.124, a trivial result would have been obtained because the overall mass-continuity equation has already been invoked through the introduction of the substantial derivative. The summation of Eq. 3.124 would simply reveal that zero equals zero. 

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. 

Derive the general vector form of the overall mass-continuity equation, recognizing that the droplet evaporation represents a source of mass to the system. 

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. 

Based on a semidifferential control volume that spans the channel radius, develop the overall and species continuity equations for flow along the tube. Show how the species equations may be written is a form that uses the substantial-derivative operator. 

Expanding the derivative term permits the incorporation of the overall mass continuity equation and the isolation of the mass-fraction derivative,... 

The derivative can be expanded to facilitate incorporation of the overall continuity equation as... 

Since there is not a continuously differentiable relationship between the inlet and outlet flows, the Gauss divergence theorem (i.e., the V- operation) has no practical application. Recall that, by definition, the surface unit vector n is directed outward. The sign of the mass-fraction difference in Eq. 16.68 is set by recognizing that the inlet flow velocity is opposite the direction of n, and vice versa for the exit. The overall mass-continuity equation,... 

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. 

The conservation equations, to be used repeatedly, are derived in Appendixes C and D and may be summarized as follows. Overall continuity, equation (D-33), is... 

For steady flow overall continuity, equation (1), reduces to... 

In a one-dimensional, spherically symmetrical system, the overall continuity equation, equation (1-23), can be written as... 

The flow velocities in flame systems are such that transport processes (diffusion and thermal conduction) make appreciable contributions to the overall flows, and must be considered in the analysis of the measured profiles. Indeed, these processes are responsible for the propagation of the flame into the fresh gas supporting it, and the exponential growth zone of the shock tube experiments is replaced by an initial stage of the reaction where active centres are supplied by diffusion from more reacted mixture sightly further downstream. The measured profiles are related to the kinetic reaction rates by means of the continuity equations governing the one-dimensional flowing system. Let Wi represent the concentration (g. cm" ) of any quantity i at distance y and time t, and let F,- represent the overall flux of the quantity (g. cm". sec ). Then continuity considerations require that the sum of the first distance derivative of the flux term and the first time derivative of the concentration term be equal to the mass chemical rate of formation q,- of the quantity, i.e. 

The interpretation of measured flame profiles by means of the continuity equations may be approached in one of two ways. The direct experimental approach involves the use of the measured profiles to calculate overall fluxes, reaction rates, and hence rate coefficients. Its successful application depends on the ability to measure the relevant profiles, including concentrations of intermediate products. This is not always possible. In addition, the overall fluxes in the early part of the reaction zone may involve large diffusion contributions, and these depend in turn on the slopes of the measured profiles. Thus accuracy may suffer. The lining up on the distance axis of profiles measured by different methods is also a problem, and, in quantitative terms, factor-of-two accuracy is probably about the best that may normally be expected from this approach at the position of maximum rate. Nevertheless, examination of the concentration dependence of reaction rates in flames may still provide useful preliminary information about the nature of the controlling elementary processes [119—121]. Some problems associated with flame profile measurements and their interpretation have been discussed by Dixon-Lewis and Isles [124]. Radical recombination rates in the immediate post-combustion zones of flames are capable of measurement with somewhat h her precision than above. 

The pressure correction equation is solved based on the normalized overall continuity equation. 

---