Conservation of solute mass

The constant ai is determined from the conservation of solute mass ... 

A general expression for the conservation of solute mass for a certain chemical component in a representative elementary volume of water-saturated porous medium fixed and rigid in space is (e.g. Garven, 1985 Garven and Freeze, 1984a)... 

It is often useful to non-dimensionalize a differential equation, not only because the resulting equation is generally both simpler in form and more general in its applicability, but also because it provides a convenient way of comparing the relative importance of terms in the equation (e.g., the importance of diffusive versus advective flux). Two important non-dimensional parameters which appear in the conservation of solute mass equation are the Peclet number and the Damkohler number. The Peclet number is given by... 

The conservation of chemical mass in the subsurface is mainly controlled by the transport of solutes through the water-saturated porous medium, the chemical reactions in the groundwater and the reactions between the groundwater plus solutes and the solid matrix. 

Analogies Between Mass, Heat and Momentum Transfer Fluxes A comparison of the partial differential equations for the conservation of heat, mass and momentum in a turbulent flow field (5.240), (5.241) and (5.242) shows that the equations are mathematically similar provided that the pressure term in the momentum equation is negligible [135]. If the corresponding boundary contitions are similar too, the normalized solution of these equations will have the same form. 

The solution to this laminar boundary layer problem must satisfy conservation of species mass via the mass transfer equation and conservation of overall mass via the equation of continuity. The two equations have been simplified for (1) two-dimensional axisymmetric flow in spherical coordinates, (2) negligible tangential diffusion at high-mass-transfer Peclet numbers, and (3) negligible curvature for mass flux in the radial direction at high Schmidt numbers, where the mass transfer... 

Besides equilibrium constant equations, two other types of equations are used in the systematic approach to solving equilibrium problems. The first of these is a mass balance equation, which is simply a statement of the conservation of matter. In a solution of a monoprotic weak acid, for example, the combined concentrations of the conjugate weak acid, HA, and the conjugate weak base, A , must equal the weak acid s initial concentration, Cha- ... 

Conservation of mass requires that the moles of solute initially present in one phase equal the combined moles of solute in the aqueous and organic phases after the extraction thus... 

The generalized standard addition method (GSAM) extends the analysis of mixtures to situations in which matrix effects prevent the determination of 8x and 8y using external standards.When adding a known concentration of analyte to a solution containing an unknown concentration of analyte, the concentrations usually are not additive (see question 9 in Chapter 5). Conservation of mass, however, is always obeyed. Equation 10.11 can be written in terms of moles, n, by using the relationship... 

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. 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

The Chapman-Jongnet (CJ) theory is a one-dimensional model that treats the detonation shock wave as a discontinnity with infinite reaction rate. The conservation equations for mass, momentum, and energy across the one-dimensional wave gives a unique solution for the detonation velocity (CJ velocity) and the state of combustion products immediately behind the detonation wave. Based on the CJ theory it is possible to calculate detonation velocity, detonation pressure, etc. if the gas mixtnre composition is known. The CJ theory does not require any information about the chemical reaction rate (i.e., chemical kinetics). 

Verify that the total mass of solute is conserved. 

To describe the diffusion of solutes in the rhizosphere, where concentration gradients change with time, /, as well as space, mass conservation is invoked with the spatial geometry appropriate for the cylindrical root (8) ... 

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 ... 

Compartmental soil modeling is a new concept and can apply to both modules. For the solute fate module, for example, it consists of the application of the law of pollutant mass conservation to a representative user specified soil element. The mass conservation principle is applied over a specific time step, either to the entire soil matrix or to the subelements of the matrix such as the soil-solids, the soil-moisture and the soil-air. These phases can be assumed in equilibrium at all times thus once the concentration in one phase is known, the concentration in the other phases can be calculated. Single or multiple soil compartments can be considered whereas phases and subcompartments can be interrelated (Figure 2) with transport, transformation and interactive equations. 

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 pipe flow, HEM requires solution of the equations of conservation of mass, energy, and momentum. The momentum equation is in differential form, which requires partitioning the pipe into segments and carrying out numerical integration. For constant-diameter pipe, these conservation equations are as follows ... 

Integration of the ODEs results in the concentration profiles for all reacting species as a function of the reaction time and the initial concentrations. The explicit solutions for the ODEs above (3.75) are given below (3.76). We list the equations for one concentration only. The remaining concentrations can be calculated from the closure principle, which is nothing else but the law of conservation of mass (e.g. in the first example [B]=[A]o-[A], where [A]o is the concentration of A at time zero). Only in example d) two concentrations need... 

Although the complete solution of ux at J will not be attempted at this point, it can be shown readily that the detonation velocity has a simple expression now that u2 and c2 have been shown to be equal. The conservation of mass equation is rewritten to show that... 

The fundamental basis for virtually all a prion mathematical models of air pollution is the statement of conservation of mass for each pollutant species. The formulation of a mathematical model of air poUution involves a number of basic steps, the first of which is a detaUed examination of the basis of the description of the diffusion of material released into the atmosphere. The second step requires that the form of interaction among the various physical and chemical processes be specified and tested against independent experiments. Once the appropriate mathematical descriptions have been formulated, it is necessary to implement suitable solution procedures. The final step is to assess the ability of the model to predict actual ambient concentration distributions. 

The key problem in using Eq. (3.1) is the specification of p. We ask whether we can derive an expression for p. The velocity components u, v, and w, although random, are related through conservation of mass and momentum for the flow, that is, they are governed by the stochastic Navier-Stokes and continuity equations. In general, as we have noted, an exact solution for u, v, and w is unobtainable. We can, however, consider an idealized situation in which the statistical properties of u, v, and w are specified a priori. Then, in so doing, we wish to see if we can obatin an exact solution of Eq. (2.4) from which p can be obtained through Eq. (2.6). 

Because water is a universal solvent, at least some of virtually every element is present as a solute in seawater. As shown in Table 3.1, the most abundant substances in seawater are the major ions (Cl , Na", SO4 , Mg ", Ca ", and K" ). They are present in nearly constant proportions in the open ocean because their concentrations are largely controlled by physical processes associated with water movement, such as transport by currents, mixing via turbulence, evaporation, and rainfall. These solutes are also referred to as conservative ions. Most of the rest of the solutes in seawater are not present in constant proportions because their concentrations are altered by chemical reactions that occur faster than the physical processes responsible for water movement. These chemicals are said to be nonconservative. Though most substances in seawater are nonconservative, they collectively comprise only a small fraction of the total mass of solutes and solids in the ocean. 

Since the ORR is a first-order reaction following Tafel kinetics, the solution of the mass conservation equation (eq 23) in a spherical agglomerate yields an analytic expression for the effectiveness factor... 

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