Mass continuity equation

Computational fluid dynamics (CFD) is the analysis of systems involving fluid flow, energy transfer, and associated phenomena such as combustion and chemical reactions by means of computer-based simulation. CFD codes numerically solve the mass-continuity equation over a specific domain set by the user. The technique is very powerful and covers a wide range of industrial applications. Examples in the field of chemical engineering are ... 

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-... 

Regardless of how an incompressible element of fluid changes shape, its volume cannot change. Therefore, for an incompressible fluid, it is apparent that volumetric dilatation must be zero. Thus it must be the case that V-V = 0 for incompressible flows. The fact that V V = 0 for an incompressible fluid is also apparent from the mass-continuity equation, Eq. 2.35. 

Ironically, since the mass-continuity equation was already used in the derivation of the substantial-derivative form of Eq. 3.2, it is not directly useful for deriving the continuity equation itself. Its application simply returns a trivial identity. Instead, we begin with the integral form as stated in Eq. 2.30 to yield... 

The assumption of incompressibility, insofar as the Navier-Stokes equations are concerned, relates to the behavior of the mass-continuity equation (Section 3.1). Consider the behavior of the steady-state continuity equation in the form... 

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. 

Up to now, the mass-continuity equations (e.g., Eq. 3.124) have been written in terms of the mass-flux vector j, which is a function of the species composition field. As noted in Section 3.5.2, different levels of theory can used to specify the functional relationship between flux and composition gradients, and mass flux can also depend on temperature or... 

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. 

Based on the spherical control volume shown in Fig 3.15, derive the mass-continuity equation. Begin with the general statement of the Reynolds transport theorm in integral form (Eq. 2.19)... 

Using the integrals in the previous questions, derive a differential-equation form of the mass-continuity equation. For the differential control volume, explain how and why the volume integrals are eliminated from the analysis. 

For deriving the mass-continuity equation, consider the general relationship between the system and the control volume ... 

Show that this relationship leads to a trivial identity, thus failing to be directly useful in deriving the mass-continuity equation. Explain the root cause of this failure. 

Write the mass-continuity equation that is appropriate for an incompressible flow. 

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. 

For long rod-guide systems it is reasonable to assume that the only nonzero velocity component is u, the axial velocity. Inasmuch as the rod and guide may have different axial velocities, it is clear that the fluid velocity must be permitted to vary radially. Given that the radial and circumferential velocities v and w are zero, the mass-continuity equation, Eq. 6.3, requires that... 

For the steady flow of an incompressible fluid, state the appropriate mass-continuity equation in spherical coordinates. What can be inferred from the reduced continuity equation about the functional form of of the circumferential velocity v 2... 

Beginning with a mass-conservation law, the Reynolds transport theorem, and a differential control volume (Fig. 4.30), derive a steady-state mass-continuity equation for the mean circumferential velocity W in the annular shroud. Remember that the pressure p 6) (and hence the density p(6) and velocity V(6)) are functions of 6 in the annulus. 

Write out the mass-continuity equation that describes the flow of an incompressible fluid in the toroidal channel. 

Deriving the mass-continuity equation begins with a mass-conservation principle and the Reynolds transport theorem. Unlike the channel with chemically inert walls, when surface chemistry is included the mass-conservation law for the system may have a source term,... 

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

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,... 

The purpose of this appendix is to spell out explicitly the Navier-Stokes and mass-continuity equations in different coordinate systems. Although the equations can be expanded from the general vector forms, dealing with the stress tensor T usually makes the expansion tedious. Expansion of the scalar equations (e.g., species or energy) are much less trouble. 

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. 

Method of solution. A trial and error method was used to solve the mass continuity equations for one of the species (e.g. CO) in the single pellet balances. To do this, expressions for other species in terms of yco are derived through the water gas-shift reaction and the reactions at the interface, i.e ... 

This is the mass continuity equation for the boundary layer. 

Atmospheric GCMs simulate the time evolution of various atmospheric fields (wind speed, temperature, surface pressure, and specific humidity), discretized over the globe, through the integration of the basic physical equations the hydrostatic equation of motion, the thermodynamic equation of state, the mass continuity equation, and a water vapor transport equation. To reproduce the... 

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