Total mass conservation Continuity equation

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... 

We first derive the so-called continuity equation, which is a direct consequence of the conservation of mass. If p is the density, or mass per unit volume, then the total mass of a fluid contained in F is equal to M = fj p dF. Letting dS — fi dS be an element of the surface, with n a unit vector perpendicular to the surface, the mass flow per unit time through the surface element is pv dS. The total fluid flow out of the volume F is then given by... 

A. TOTAL CONTINUITY EQUATION (MASS BALANCE). The principle of the conservation of mass when applied to a dynamie system says... 

We notice that the total mass is conserved because the mass density p = maua + wjb b obeys the continuity equation... 

The physical significance of these boundary conditions is as follows. Equation (26a) represents the fact that just before the tracer is injected into the system the concentration is everywhere zero. Equations (26b) and (26g) are obvious since a finite amount of tracer is injected. Equations (26d) and (26e) follow from conservation of mass at the boundaries between the sections (W4) the total mass flux entering the boundary must equal that leaving. Equations (26c) and (26f) are based on the physically intuitive argument that concentration should be continuous in the neighborhood of any point. These boundary conditions will be used extensively in the subsequent derivations. 

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. 

There is of course an equation for each species, subject to conservation of total mass given by the continuily equation, and we implicitly used conservation of total mass as stoichiometric constraints, Njo — Nj)lvj are equal for all 7 in a batch (closed) system or (Fjo — Fj)/vj are equal for aU j in a steady-state continuous reactor. 

This equation can be applied to the total mass involved in a process or to a particular species, on either a mole or mass basis. The conservation law for mass can be applied to steady-state or unsteady-state processes and to batch or continuous systems. A steady-state system is one in which there is no change in conditions (e.g., temperature, pressure) or rates of flow with time at any given point in the system the accumulation term then becomes zero. If there is no chemical reaction, the generation term is zero. All other processes are classified as unsteady state. 

Our first major task is the description of the interfacial mass transfer process and, therefore, we shall examine further the equations for continuity of species i and the equation for conservation of total mass of mixture. 

Although the derivation of the continuity equation by use of a fixed control volume is perfectly satisfactory, it is less obvious how to apply Newton s laws of mechanics in this framework. The familiar use of these principles from coursework in classical mechanics is that they are applied to describe the motion of a specific body subject to various forces or torques. To apply these same laws to a fluid (i.e., a liquid or a gas), we introduce the concepts of material points and a material volume (or material control volume) that we denote as Vm(t). Now a material point is a continuum point that moves with the local continuum velocity of the fluid. A material volume Vm (t), is a macroscopic control volume whose shape at some initial instant, / = 0, is arbitrary, that contains a fixed set of material points. Because the material volume contains a fixed set of such points, it must move with the local continuum velocity of the fluid at every point. Hence, as illustrated in Fig. 2-3, it must deform and change volume in such a way that the local flux of mass through all points on its surface is identically zero for all time (though, of course, there may still be exchange of molecules due to random molecular motion). Because mass is neither created nor destroyed according to the principle of mass conservation, the total mass contained... 

Here, the symbol D/Dt stands for the convected or material time derivative, which we shall subsequently discuss in some detail. In the context of (2-6) it is clear that D/Dt represents the time derivative of the total mass of material in the material volume Vm(t). Alternatively, we could say that it is the time derivative of the total mass associated with the fixed set of material points that comprise Vm(t). We shall see shortly that Eq. (2 6), which derives directly from the definition of a material volume for a fluid that conserves mass, is entirely equivalent to (2-3) or (2 4) and leads precisely to the pointwise continuity equation, (2-5). However, this cannot be seen easily without further discussion of the convected or material time derivative. 

Because the initial choice of Vm(t) is arbitrary, we obtain the same differential form for the continuity equation, (2-5), that we derived earlier by using a fixed control volume. Of course, the fact that we obtain the same form for the continuity equation is not surprising. The two derivations are entirely equivalent. In the first, conservation of mass is imposed by the requirement that the time rate of change of mass in a fixed control volume be exactly balanced by a net imbalance in the influx and efflux of mass through the surface. In particular, no mass is created or destroyed. In the second approach, we define the material volume element so that the mass flux through its surface is everywhere equal to zero. In this case, the condition that mass is conserved means that the total mass in the material volume element is constant. The differential form (2-5) of the statement of mass conservation, which we have called the continuity equation, is the main result of this section. 

From the definitions given it is clear that Yj = Z MjC/Uj - u) = 0, while X/ = 0. due to the conservation of mass in a reacting system. So, if each term of Eq. 7.2.a-l is multiplied by the molecular weight Mj, and the equation is then summed over the total number of species N, accounting for the relation Pf — Yj MjCj, the total continuity equation is obtained ... 

The conservation of mass is mathematically described by the continuity equation. Its physical interpretation means that a mass, represented by the total mass flow rate, entering a fluid element must be equal to the total mass flow rate leaving it. In other words, the total mass flow out of a fluid element must be equal to the time rate of total mass decrease in the fluid element as... 

Imagine that we start with a set of identical systems, whose states are distributed in phase space according to a density distribution p(P) at time / = 0, and let the systems move according to their equations of motion. The ensemble constituted by the systems (points in phase space) evolves in time. As the systems evolve, the density distribution p(P) should, in general, change with time. However, systems just move, no new systems are created, and none of the systems is destroyed. Therefore, there should be a conservation law for the probability density, similar to the continuity equation (mass conservation) of hydrodynamics. The conservation law in phase space is called the Liouville theorem. It states that the total time... 

A similar equation can be written for component 2. In addition the continuity equation (6.21) must be written and so the two conservation equations are not independent. If it can be assumed that the total concentration in the fluid phase c remains constant, which is likely to be true for gaseous systems in which the pressure drop is small, and approximately true for liquid mixtures when the components have similar molar volumes, then the mass conservation equations can be combined. If it is possible to neglect axial dispersion, then the propagation velocity of points of given concentration in the MTZ is given by ... 

Note that the Liouville equation, formally, is identical with the first conservation equation, the so-called continuity equation of hydrodynamics, equation (la). The change of the mass density and the change of the phase-space-distribution can be derived based on the conservation of the total mass and the total number of systems, respectively.) The last step of equation (7) is a definition of the term A(/ ) called the phase-space compression factor. In the case of conservative systems (the most common example of which is Hamilton s equations), the Liouville equation describes an incompressible flow and the right-hand side of equation (7) is zero. (In many statistical mechanical texts, only this incompressible form is referred to as the Liouville equation.)... 

The species continuity equation (CE) is an expression of the Lavoisier general law of conservation of mass. Equation 2.1 presents the CE in vector form and provides the proper context for the various types of chemical mass transport processes needed for chemical modeling and fate analysis. In Section 2.2.2, the mass accumulation portion of the CE is highlighted as the principal term for assessing chemical fate in the media compartments. This term includes reaction, advection, diffusion, and turbulent transport and dispersion processes. Because the magnitude and direction of this term reflect the sum total of all processes, this term uniquely defines chemical fate. In Equation 2.2, the steady-state CE minus the reaction term is commonly referred to as the advective-diffusive (AD) equation. It provides the appropriate starting point for addressing the various transport processes associated with the mobile phases in near-surface soils. 

The species continuity equations ensure conservation of atoms by balancing the chemical equations governing the reaction system. The concentration of an individual species i will be expressed either as a product of total molar density and mole fraction X, or as a product of density and mass fraction w,. In an N-component system mass fraction and mole fraction are related by... 

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. 

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