Mass Continuity

Regardless of what other conservation equations may be appropriate, a bulk-fluid mass-conservation equation is invariably required in any fluid-flow situation. When N is the mass m, the associated intensive variable (extensive variable per unit mass) is r) = 1. That is, r) is the mass per unit mass is unity. For the circumstances considered here, there is no mass created or destroyed within a control volume. Chemical reaction, for example, may produce or consume individual species, but overall no mass is created or destroyed. Furthermore the only way that net mass can be transported across the control surfaces is by convection. While individual species may diffuse across the control surfaces by molecular actions, there can be no net transport by such processes. This fact will be developed in much depth in subsequent sections where mass transport is discussed. 

We note in passing that two-phase flow leads to circumstances where for a given phase there are source or sink terms. For example, consider situation wherein water droplets are evaporating in a moist-air flow. It is possible to write mass-conservation equations for the liquid and the vapor phases. The conversion of liquid to vapor (and vice-versa) causes 

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 

Assuming a vanishingly small differential control volume, the integrand can be assumed to be uniform over the volume. Therefore the continuity equation can be written in differential-equation form as 

While the continuity equation can certainly be used in this form, it is more common to introduce the substantial derivative of density. Using the definition of the substantial derivative, which is stated as 

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In a 500 ml. bolt-head flask, provided with a mechanical stirrer, place 70 ml. of oleum (20 per cent. SO3) and heat it in an oil bath to 70°. By means of a separatory funnel, supported so that the stem is just above the surface of the acid, introduce 41 g. (34 ml.) of nitrobenzene slowly and at such a rate that the temperature of the well-stirred mixture does not rise above 100-105°. When all the nitrobenzene has been introduced, continue the heating at 110-115° for 30 minutes. Remove a test portion and add it to the excess of water. If the odour of nitrobenzene is still apparent, add a further 10 ml. of fuming sulphuric acid, and heat at 110-115° for 15 minutes the reaction mixture should then be free from nitrobenzene. Allow the mixture to cool and pour it with good mechanical stirring on to 200 g. of finely-crushed ice contained in a beaker. AU the nitrobenzenesulphonic acid passes into solution if a little sulphone is present, remove this by filtration. Stir the solution mechanically and add 70 g. of sodium chloride in small portions the sodium salt of m-nitro-benzenesulphonic acid separates as a pasty mass. Continue the stirring for about 30 minutes, allow to stand overnight, filter and press the cake well. The latter will retain sufficient acid to render unnecessary the addition of acid in the subsequent reduction with iron. Spread upon filter paper to dry partially. 

Operational Characteristics. Oxygen generation from chlorate candles is exothermic and management of the heat released is a function of design of the total unit iato which the candle is iacorporated. Because of the low heat content of the evolved gas, the gas exit temperature usually is less than ca 93°C. Some of the heat is taken up within the candle mass by specific heat or heat of fusion of the sodium chloride. The reacted candle mass continues to evolve heat after reaction ends. The heat release duting reaction is primarily a function of the fuel type and content, but averages 3.7 MJ/m (100 Btu/fT) of evolved oxygen at STP for 4—8 wt % iron compositions. 

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-... 

As nephron mass continues to decline, the sodium load overwhelms the remaining nephrons and total sodium excretion is decreased, despite the increase in sodium excretion by... 

High-frequency pressure oscillations (10-100 Hz) related to time required for pressure wave propagation in system Low-frequency oscillations (-1 Hz) related to transit time of a mass-continuity wave Occurs close to film boiling... 

Piping systems often involve interconnected segments in various combinations of series and/or parallel arrangements. The principles required to analyze such systems are the same as those have used for other systems, e.g., the conservation of mass (continuity) and energy (Bernoulli) equations. For each pipe junction or node in the network, continuity tells us that the sum of all the flow rates into the node must equal the sum of all the flow rates out of the node. Also, the total driving force (pressure drop plus gravity head loss, plus pump head) between any two nodes is related to the flow rate and friction loss by the Bernoulli equation applied between the two nodes. 

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. 

It should be noted that equation 7.25 represents the interaction of forces acting on bubbles in a swarm while equation 7.24 represents the principle of conservation of mass (continuity). Both equations must be satisfied simultaneously. 

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. 

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... 

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