The Conservation of Mass

The law of conservation of mass is a statement of the mass balance for flow in and out and changes of mass storage of a system. Change in mass caused by any energy transfer such as in a chemical reaction or a combustion process is negligibly small and therefore is not taken into account. For analysis purposes, the conservation of mass law is presented for both the system and the control volume. 

Since a system is defined as a fixed and identifiable quantity of mass, the conservation mass for a system is defined as 

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Count the number of species whose concentrations appear in the equilibrium constant expressions these are your unknowns. If the number of unknowns equals the number of equilibrium constant expressions, then you have enough information to solve the problem. If not, additional equations based on the conservation of mass and charge must be written. Continue to add equations until you have the same number of equations as you have unknowns. 

In an indirect volatilization gravimetric analysis, the change in the sample s weight is proportional to the amount of analyte. Note that in the following example it is not necessary to apply the conservation of mass to relate the analytical signal to the analyte. 

Conservation of Mass. The general equations for the conservation of mass are the scalar equations (Fig. 21a) ... 

The conservation of mass gives comparatively Httle useful information until it is combined with the results of the momentum and energy balances. Conservation of Momentum. The general equation for the conservation of momentum is... 

Deterministic air quaUty models describe in a fundamental manner the individual processes that affect the evolution of pollutant concentrations. These models are based on solving the atmospheric diffusion —reaction equation, which is in essence the conservation-of-mass principle for each pollutant species... 

Dynamic meteorological models, much like air pollution models, strive to describe the physics and thermodynamics of atmospheric motions as accurately as is feasible. Besides being used in conjunction with air quaHty models, they ate also used for weather forecasting. Like air quaHty models, dynamic meteorological models solve a set of partial differential equations (also called primitive equations). This set of equations, which ate fundamental to the fluid mechanics of the atmosphere, ate referred to as the Navier-Stokes equations, and describe the conservation of mass and momentum. They ate combined with equations describing energy conservation and thermodynamics in a moving fluid (72) ... 

As a reactant molecule from the fluid phase surrounding the particle enters the pore stmcture, it can either react on the surface or continue diffusing toward the center of the particle. A quantitative model of the process is developed by writing a differential equation for the conservation of mass of the reactant diffusing into the particle. At steady state, the rate of diffusion of the reactant into a shell of infinitesimal thickness minus the rate of diffusion out of the shell is equal to the rate of consumption of the reactant in the shell by chemical reaction. Solving the equation leads to a result that shows how the rate of the catalytic reaction is influenced by the interplay of the transport, which is characterized by the effective diffusion coefficient of the reactant in the pores, and the reaction, which is characterized by the first-order reaction rate constant. 

The conservation of mass law finds a major application during the performance of pollution-prevention assessments. As described earlier, a pollution-prevention assessment is a systematic, planned procedure with the objective of identifying methods to reduce or ehminate waste. The assessment process should characterize the selected waste streams and processes (Ref. 11)—a necessaiy ingredient if a material balance is to be performed. Some of the data required for the material balance calciilation may be collected during the first review of site-specific data however, in some instances, the information may not be collected until an actual site walk-through is performed. 

Fundamental Equations A complete development of the fundamental equations is presented elsewhere (Growl aud Louvar, 1990, pp. 129-144). The model begins by writing an equation for the conservation of mass of the dispersing material ... 

Equations (2.45), (2.46), and (2.49) express the conservation of mass, momentum, and energy in Lagrangian coordinates for continuous flow. 

To demonstrate that the Rayleigh line actually represents the thermodynamic path to which material is subjected on being shocked from state p = 0, F = Fq to P = Pi, F = Fi, we demonstrate below that the shock wave sketched in Fig. 4.1 must be steady. Moreover, the Rankine Hugoniot equations ((4.1)-(4.3)) not only describe the conservation of mass, momentum. 

The conservation equations are more commonly written in the initial reference frame (Lagrangian forms). The time derivative normally used is d /dt. Equation (9.5) is used to derive (9.2) from the Lagrangian form of the conservation of mass... 

The objectives are not realized when physical modeling are applied to complex processes. However, consideration of the appropriate differential equations at steady state for the conservation of mass, momentum, and thermal energy has resulted in various dimensionless groups. These groups must be equal for both the model and the prototype for complete similarity to exist on scale-up. 

Elementary single-component systems are those that have just one chemical species or material involved in the process. Filling of a vessel is an example of this kind. The component can be a solid liquid or gas. Regardless of the phase of the component, the time dependence of the process is captured by the same statement of the conservation of mass within a well-defined region of space that we will refer to as the control volume. 

In this chapter we will apply the conservation of mass principle to a number of different kinds of systems. While the systems are different, by the process of analysis they will each be reduced to their most common features and we will find that they are more the same than they are different. When we have completed this chapter, you will understand the concept of a control volume and the conservation of mass, and you will be able to write and solve total material balances for single-component systems. 

The conserved quantities that are of utmost importance to a chemical engineer are mass, energy, and momentum. It is the objective of this text to teach you how to utilize the conservation of mass in the analysis of units and processes that involve mass flow and transfer and chemical reaction. For each conserved quantity the principle is the same—conserved quantities are... 

Therefore, iiweknowthecompactedbulkdensity,thenitispossibletocomputethemass in thebed using the mathematical statement for the conservation of mass. In this case the reactoranditsphysical dimensions define thecontrol volume. The rateofcatalyst delivery is a constant that we will call min The rate of mass flow out of the reactor is zero, that is,... 

To find the concentration at the point where the streams comeback together we again apply the conservation of mass. The mass of dye in the two lines coming into this mixing pointperunittimemustbeequaltothetotalmassgoingoutofitperunittime.Therateof mass flowinisjustthesumoftheproductsoftheconcentrationsand flowratesofthetwo... 

This chapter sets out to provide a means of handling these types of interphase mass transfer problems taking into consideration their fundamental characterizing variables, the conservation of mass, and appropriate constitutive relationships. 

The model equations are determined by writing the balance equations based on the conservation of mass and energy. Tlie balance equations have the following basic form ... 

Computational fluid dynamics (CFD) is the numerical analysis of systems involving transport processes and solution by computer simulation. An early application of CFD (FLUENT) to predict flow within cooling crystallizers was made by Brown and Boysan (1987). Elementary equations that describe the conservation of mass, momentum and energy for fluid flow or heat transfer are solved for a number of sub regions of the flow field (Versteeg and Malalase-kera, 1995). Various commercial concerns provide ready-to-use CFD codes to perform this task and usually offer a choice of solution methods, model equations (for example turbulence models of turbulent flow) and visualization tools, as reviewed by Zauner (1999) below. 

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... 

Using the laws of constant composition and the conservation of mass, complete the molecular picture of hydrogen molecules (O—O) reacting with chlorine molecules ( — ) to give hydrogen chloride ( —O) molecules. 

Since the mass of a mole of water is the sum of the masses of the atoms in the mole of water, the conservation of mass implies conservation of atoms. 

Al(III) is an example of an aquatic ion that forms a series of hydrated and protonated species. These include AlOrf Al(OH)J, Al(OH)3, and other forms in addition to AP. (For simplicity, we omit the H2O molecules that complete the structures of these complexes.) Most of these species are amphoteric (able to act as an acid or a base). Thus the speciation of Al(III) and many other aquatic ions is sensitive to pH. In this case, an aggregate variable springs from the conservation of mass condition. In the case of dissolved aluminum, the total dissolved aluminum is given by... 

There are two levels, discrete particle level and continuum level, for describing and modeling of the macroscopic behaviors of dilute and condensed matters. The physics laws concerning the conservation of mass, momentum, and energy in motion, are common to both levels. For simple dilute gases, the Boltzmann equation, as shown below, provides the governing equation of gas dynamics on the discrete particle level... 

One of the basic principles of modelling is that of the conservation of mass. For a steady-state flow process, this can be expressed by the statement ... 

The previous discussion has been in terms of the total mass of the system, but most process streams, encountered in practice, contain more than one chemical species. Provided no chemical change occurs, the generalised dynamic equation for the conservation of mass can also be applied to each chemical component of the system. Thus for any particular component... 

Viscometric flow theories describe how to extract material properties from macroscopic measurements, which are integrated quantities such as the torque or volume flow rate. For example, in pipe flow, the standard measurements are the volume flow rate and the pressure drop. The fundamental difference with spatially resolved measurements is that the local characteristics of the flows are exploited. Here we focus on one such example, steady, pressure driven flow through a tube of circular cross section. The standard assumptions are made, namely, that the flow is uni-directional and axisymmetric, with the axial component of velocity depending on the radius only. The conservation of mass is satisfied exactly and the z component of the conservation of linear momentum reduces to... 

The modeling of mass transport in packed bed reactors applies the theory of dispersion [32]. The conservation of mass for the average concentration [Pg.515]

Vessel blowdown. The previously mentioned relationships for the critical flow rate of a steam-water mixture can be employed with the conservation of mass and energy for a vessel of fixed volume to determine its time-dependent blowdown properties. The range of problems associated with coolant decompression in water-cooled reactors is quite broad. The types of hypothetical (some are even incredible) reactor accidents may be... 

Let us now interrelate the fluxes in a variety of meaningful ways. All fluxes have units of moles per square centimeter per second. The conservation of mass is satisfied by the requirement that the disappearance rate of drug from the solution is equal to the sum of fluxes of drug and metabolite emerging from the cell ... 

The material balances ensure the conservation of mass around each unit and concerning each state involved in production scheduling. 

As discussed in Chapter 1, the basic principles that apply to the analysis and solution of flow problems include the conservation of mass, energy, and momentum in addition to appropriate transport relations for these conserved quantities. For flow problems, these conservation laws are applied to a system, which is defined as any clearly specified region or volume of fluid with either macroscopic or microscopic dimensions (this is also sometimes referred to as a control volume ), as illustrated in Fig. 5-1. The general conservation law is... 

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... 

The conservation of mass can be applied to an arbitrarily small fluid element to derive the microscopic continuity equation, which must be satisfied at all points within any continuous fluid. This can be done by considering an arbitrary (cubical) differential element of dimensions dx, dy, dz, with mass... 

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-... 

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