Mass conservation, continuity

The quantities p, p, and Ux represent the distiubance associated with the propagation of sound. Here, we consider the fluid at rest (i.e. Uo = 0) as the undisturbed state, with pressure po and density po. Although this case differs from the one explained in Figure 6.1, where the fluid is flowing in its undisturbed state, sound propagation is more basically introduced in this simpler case, and the other case of the water hammer phenomenon will then be easily treated. The space and time variations of the three variables are hnked by mass conservation (continuity equation), the velocity component along Ox (Euler s equation), and compressibility (Chapter 1, Table 1.1). 

Level of enforcement of the incompressibility condition depends on the magnitude of the penalty parameter. If this parameter is chosen to be excessively large then the working equations of the scheme will be dominated by the incompressibility constraint and may become singular. On the other hand, if the selected penalty parameter is too small then the mass conservation will not be assured. In non-Newtonian flow problems, where shear-dependent viscosity varies locally, to enforce the continuity at the right level it is necessary to maintain a balance between the viscosity and the penalty parameter. To achieve this the penalty parameter should be related to the viscosity as A = Xorj (Nakazawa et al, 1982) where Ao is a large dimensionless parameter and tj is the local viscosity. The recommended value for Ao in typical polymer flow problems is about 10. ... 

Transport of component i in a binary system is described by the equation of continuity [2], which is an expression for mass conservation of the subject component in the system, i.e.,... 

Semibatch or semiflow processes are among the most difficult to analyze from the viewpoint of reactor design because one must deal with an open system under nonsteady-state conditions. Hence the differential equations governing energy and mass conservation are more complex than they would be for the same reaction carried out batchwise or in a continuous flow reactor operating at steady state. 

For a given system (e.g., Fig. 5-1), each entering stream (subscript i) will carry mass into the system (at rate rh ), and each exiting stream (subscript o) carries mass out of the system (at rate rh0). Flence, the conservation of mass, or continuity, equation for the system is... 

Note that there are 11 dependent variables, or unknowns in these equations (three u s, six r,y s, P, and p), all of which may depend on space and time. (For an incompressible fluid, p is constant so there are only 10 unknowns. ) There are four conservation equations involving these unknowns (the three momentum equations plus the conservation of mass or continuity equation), which means that we still need six more equations (seven, if the fluid is compressible). These additional equations are the con-... 

The general equation used for conservation of mass (the continuity equation) may be written as follows ... 

Since the particles are randomly located in grid cells, interpolation and particle-field-estimation algorithms are required. Special care is needed to ensure local mass conservation (i.e., continuity) and to eliminate bias. 

Finally, continuity equations that account for mass conservation laws. 

This equation is usually referred to as the continuity equation or mass conservation equation. The source term, Sm, in the continuity equation is commonly caused by mass consumption or production from electrochemical reactions as well as mass loss/gain through phase transformation. 

We next have to consider the continuity equation, which students first encounter seriously in introductory chemistry and physics as the principle of mass conservation. For any fluid we require that the total mass flow into some element of volume minus the flow out is equal to the accumulation of mass, and we either write these as integral balances (stoichiometry) or as differential balances on a differential element of volume. 

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. 

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. 

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,... 

Turn now to the individual species continuity equations where the mass-conservation law for the system includes both homogeneous- and heterogeneous-chemistry source terms,... 

Conservation Laws. The fundamental conservation laws of physics can be used to obtain the basic equations of fluid motion, the equations of continuity (mass conservation), of flow (momentum conservation), of... 

Mass Balance, Continuity Equation The continuity equation, expressing conservation of mass, is written in cartesian coordinates as... 

For the total mass conservation of a single-phase fluid, / represents the fluid density p. jr represents the diffusional flux of total mass, which is zero. For flow systems without chemical reactions, d> = 0. Therefore, from Eq. (5.12), we have the continuity equation as... 

Equation 10-5 is the unsteady, three-dimensional mass conservation or continuity equation at a point in a compressible fluid. The first term on the left side is the rate of change in time of the density (mass per unit volume). The second term describes the net flow of mass leaving the element across its boundaries and is called the convective term. 

In order to determine the distributions of pressure, velocity, and temperature the principles of conservation of mass, conservation of momentum (Newton s Law) and conservation of energy (first law of Thermodynamics) are applied. These conservation principles represent empirical models of the behavior of the physical world. They do not, of course, always apply, e.g., there can be a conversion of mass into energy in some circumstances, but they are adequate for the analysis of the vast majority of engineering problems. These conservation principles lead to the so-called Continuity, Navier-Stokes and Energy equations respectively. These equations involve, beside the basic variables mentioned above, certain fluid properties, e.g., density, p viscosity, p conductivity, k and specific heat, cp. Therefore, to obtain the solution to the equations, the relations between these properties and the pressure and temperature have to be known. (Non-Newtonian fluids in which p depends on the velocity field are not considered here.) As discussed in the previous chapter, there are, however, many practical problems in which the variation of these properties across the flow field can be ignored, i.e., in which the fluid properties can be assumed to be constant in obtaining fire solution. Such solutions are termed constant... 

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