Balance species mass

The Gaussian Plume Model is the most well-known and simplest scheme to estimate atmospheric dispersion. This is a mathematical model which has been formulated on the assumption that horizontal advection is balanced by vertical and transverse turbulent diffusion and terms arising from creation of depletion of species i by various internal sources or sinks. In the wind-oriented coordinate system, the conservation of species mass equation takes the following form ... 

The Navier-Stokes equations are solved first to determine the velocity field throughout the reactor, as described by Armaou and Christofides [4], and subsequently by Brass and Lee [5] using FEMLAB. Then, the species mass balances are solved to determine the concentrations of SiFL (1), SiH2 (2), SiH3 (3), and H (4), throughout the reactor. Finally, the deposition rate of silicon is ... 

FIGURE 3.7 Species mass fraction profiles for a constant temperature reaction of moist CO oxidation. Initial conditions temperature = 1100K, pressure = latm, Xco = 0.002, XH0 = 0.01, Xq2 = 0.028, and the balance N2 where X are the initial mole fractions. 

Correct modeling of variable diffiisivity, time-dependent emission sources, nonlinear chemical reactions, and removal processes necessitates numerical integrations of the species-mass-balance equations. Because of limitations of dispersion data, emission data, or chemical rate data, this approach to the modeling of air pollution may not necessarily ensure higher fidelity, but it does hold out the possibility of the incorporation of more of these details as they become known. 

Similarity solutions of the species-mass-balance equations were assumed by Friedlander and Seinfeld for a simple photochemical-smog reaction scheme. (This scheme assumed a steady-state condition for ozone.) Demonstration runs were shown for parametric variations in the... 

Pollutants emitted by various sources entered an air parcel moving with the wind in the model proposed by Eschenroeder and Martinez. Finite-difference solutions to the species-mass-balance equations described the pollutant chemical kinetics and the upward spread through a series of vertical cells. The initial chemical mechanism consisted of 7 species participating in 13 reactions based on sm< -chamber observations. Atmospheric dispersion data from the literature were introduced to provide vertical-diffusion coefficients. Initial validity tests were conducted for a static air mass over central Los Angeles on October 23, 1968, and during an episode late in 1%8 while a special mobile laboratory was set up by Scott Research Laboratories. Curves were plotted to illustrate sensitivity to rate and emission values, and the feasibility of this prediction technique was demonstrated. Some problems of the future were ultimately identified by this work, and the method developed has been applied to several environmental impact studies (see, for example, Wayne et al. ). 

We need reaction-rate expressions to insert into species mass-balance equations for a particular reactor. These are the equations from which we can obtain compositions and other quantities that we need to describe a chemical process. In introductory chemistry courses students are introduced to first-order irreversible reactions in the batch reactor, and the impression is sometimes left that this is the only mass balance that is important in chemical reactions. In practical situations the mass balance becomes more comphcated. 

In this chapter we consider how we should design chemical reactors when we want to produce a specific product while converting most of the reactant and rninitnizing the production of undesired byproducts. It is clear that in order to design any chemical process, we need to be able to formulate and solve the species mass-balance equations in multiple-reaction systems to determine how we can convert reactants into valuable products efficiently and economically. 

We could write species mass-balance equations (S = 6 in this example) on any such reaction sequence and solve these (/ = 4 are inseparable) to find Cj x), and in most practical examples we must do this. However, there are two simple reaction networks that provide insight into these more complex networks, and we wiU next consider them, namely, series and parallel reaction networks (Figure 4-3). 

For the CSTR we have S algebraic species mass-balance equations. We can eliminate S R of these to obtain R irreducible algebraic polynomials in R of the S species which we... 

We are also concerned with gradients in composition throughout the reactor. We have thus far been concerned only with the very small gradient dCj/dz down the reactor from inlet to exit, which we encounter in the species mass balance, which we must ultimately solve. Then there is the gradient in Cj around the catalyst pellet Finally, there is the gradient within the porous catalyst pellet and around the catalytic reaction site within the pellet As we consider... 

These equations apply to the total mass or mass density of the system, while we use moles when describing chemical reaction. Therefore, whenever we need to solve these equations simultaneously, we must transform our species mass balances into weight fraction when including momentum and total mass-balance equations. 

Fine Aerosol Species Mass Balance, China Lake, California... 

The analysis of polymer processing is reduced to the balance equations, mass or continuity, energy, momentum and species and to some constitutive equations such as viscosity models, thermal conductivity models, etc. Our main interest is to solve this coupled nonlinear system of equations as accurately as possible with the least amount of computational effort. In order to do this, we simplify the geometry, we apply boundary and initial conditions, we make some physical simplifications and finally we chose an appropriate constitutive equations for the problem. At the end, we will arrive at a mathematical formulation for the problem represented by a certain function, say / (x, T, p, u,...), valid for a domain V. Due to the fact that it is impossible to obtain an exact solution over the entire domain, we must introduce discretization, for example, a grid. The grid is just a domain partition, such as points for finite difference methods, or elements for finite elements. Independent of whether the domain is divided into elements or points, the solution of the problem is always reduced to a discreet solution of the problem variables at the points or nodal pointsinxxnodes. The choice of grid, i.e., type of element, number of points or nodes, directly affects the solution of the problem. 

HI) The mass balance is required for each species. Mass balance for each phase is obtained via mass balances of the species it contains. 

Table A.2 is model output for seawater freezing at 253.15 K. Beneath the title, the output includes temperature, ionic strength, density of the solution (p), osmotic coefficient amount of unfrozen water, amount of ice, and pressure on the system. Beneath this line are the solution and gaseous species in the system. The seven columns include species identification, initial concentration, final (equilibrium) concentration, activity coefficient, activity, moles in the solution phase, and mass balance. The mass balance column only contains those components for which a mass balance is maintained. The number of these components minus 1 is generally the number of independent components in the system (in this case, 8 — 1 = 7). The mass balances (col. 7) should equal the initial concentrations (col. 2). This mass balance comparison is a good check on the computational accuracy.

The equation of the ADF model flow can be obtained by making a particular species mass balance, as in the case of a plug flow model. In this case, for the beginning of species balance we must consider the axial dispersion perturbations superposed over the plug flow as shown in Fig. 3.31. In the description given below, the transport vector has been divided into its convective and dispersion components. 

The procedure is therefore to write a K2Cr04 balance, solve it simultaneously with Equation 1 to determine ttu and 7725, and then write a total mass balance to determine m2. The additive terms in each equation have units of kg/h of the balanced species. 

Write an expression for the total amount of the balanced species in the system [K(m )p(kg/m ) for total mass, V(m )CA(mol A/m ) or n,otai(mol)xA(mol A/mol) for species A], Differentiate the expression with respect to time to obtain the accumulation term in the balance equation. 

The TLM (Davis and Leckie, 1978) is the most complex model described in Figure 4. It is an example of an SCM. These models describe sorption within a framework similar to that used to describe reactions between metals and ligands in solutions (Kentef fll., 1988 Davis and Kent, 1990 Stumm, 1992). Reactions involving surface sites and solution species are postulated based on experimental data and theoretical principles. Mass balance, charge balance, and mass action laws are used to predict sorption as a function of solution chemistry. Different SCMs incorporate different assumptions about the nature of the solid - solution interface. These include the number of distinct surface planes where cations and anions can attach (double layer versus triple layer) and the relations between surface charge, electrical capacitance, and activity coefficients of surface species. 

The Mathematical Model Once a conceptual PBPK model has been created, with some knowledge of the chemical of interest, this representation can be translated into mathematical equations for use in predicting time-course disposition. The fundamental equations utilized arise from chemical species mass balances, which account for the rates at which molecules enter and leave each compartment, as well as other processes (e.g., rates of reactions) that produce or consume the chemical. 

A chemical plant includes tens to hundreds of process units, such as chemical reactors, heat exchangers, distillation columns, absorption towers, etc. For each unit, material and energy balances are used to relate input and output streams. Rate equations and equilibrium relations help describe the conversion of species, mass, and energy in the units. Collectively, these equations provide the equality constraints for the plant model. 

In general problems in solution chemistry there is also a fourth kind of equation called a mass-balance equation. Mass-balance expressions relate the total concentration of a species reported in the chemical analysis to concentrations of the several forms of that species in solution. For example,... 

It turns out, however, that we will not need this or any other mass-balance equation to solve our rain pH problem. For rain, we know the pH is generally below about 6, so that mOH and mCO " are negligible relative to mH" and mHCOj. As a rule of thumb, any species whose concentration is less than 1% (two orders of magnitude) of another species concentration present is negligible in a charge-balance or mass-balance calculation. (How do we know these two species are negligible at this pH ) With this simplification, and ignoring activity coefficients in such a dilute water, we find expression (8.8) reduces to simply mH = mHCOj. 

For an individual species c in a multi-component mixture the species mass balance principle postulates that the time rate of change of the mass of a system of species c is equal to the rate at which the mass of c is produced by homogeneous chemical reactions. 

Let Vc t) denote the material volume occupied by species c (see Fig. I.IB), and Rc denote the rate of production of species c per unit volume by homogeneous chemical reactions. The species mass balance can thus be expressed as ... 

The transport theorem (1.36) enables the time rate of change in the species mass balance (1.35) to be transformed into the Eulerian formulation. Hence,... 

Equation (1.46) must be satisfied for any macroscopic volume V)., thus the expression inside the volume integral must be equal to zero. The resulting differential species mass balance coincides with (1.39). 

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