Species balance equations

Our discussion of multiphase CFD models has thus far focused on describing the mass and momentum balances for each phase. In applications to chemical reactors, we will frequently need to include chemical species and enthalpy balances. As mentioned previously, the multifluid models do not resolve the interfaces between phases and models based on correlations will be needed to close the interphase mass- and heat-transfer terms. To keep the notation simple, we will consider only a two-phase gas-solid system with ag + as = 1. If we denote the mass fractions of Nsp chemical species in each phase by Yga and Ysa, respectively, we can write the species balance equations as... 

In principle, one can carry out a four-dimensional optimization in which the four parameters are varied subject to constraints (< 1 and P4 < 1 ), to minimize the deposition time with the non-uniformity bounded e.g., MN < 3. However, objective function evaluations involve solutions of the Navier-Stokes and species balance equations and are computationally expensive. Instead, Brass and Lee carry out successive unidirectional optimizations, which show the key trends and lead to excellent designs. A summary of the observed trends is shown in Table 10.4-1. Both the deposition rate and the non-uniformity are monotonic functions of the geometric parameters within the bounds considered, with the exception that the non-uniformity goes through a minimum at optimal values of P3 and P4. 

In another review, Hoffert discussed the social motivations for modeling air quality for predictive purposes and elucidated the components of a model. Meteorologic factors were summarized in terms of windfields and atmospheric stability as they are traditionally represented mathematically. The species-balance equation was discussed, and several solutions of the equation for constant-diffusion coefficient and concentrated sources were suggested. Gaussian plume and puff results were related to the problems of developing multiple-source urban-dispersion models. Numerical solutions and box models were then considered. The review concluded with a brief outline of the atmospheric chemical effects that influence the concentration of pollutants by transformation. 

The derivation of the mixture-balance laws has been given by Chapman and Cowling for a binary mixture. Its generalization to multicomponent mixtures, as in Equation 5-1, uses a determination of the invariance of the Boltzmann equation. This development has been detailed by Hirschfelderet These derivations were summarized in the notes of Theodore von Karmin s Sorbonne lectures given in 1951-1952, and the results of his summaries were stated in Pinner s monograph. For turbulent flow, the species-balance equation can be represented in the Boussinesq approximation as ... 

The complete (but still containing many approximations) species balance equations are... 

Mole balance expressions were developed for a general series reaction by Agarwalla and Lund [16], and the same procedures were used here to develop the species balance equations shown in Table I. Boundary conditions and parameter definitions are presented in Tables II and III. Note that the boundary conditions are given only for co-current flow of reactants and inert, which is the only configuration studied. Previous work [16], has shown that counter-current operation is less effective than co-current operation. 

Since the fiber phase is not stationary, the surface integral cannot be set to zero without further considerations. As shown earlier, dBr/dt = 1/V js Ur hids (see Eq. 5.10). Because der/dt = 0 in the IP process, the contribution of the surface integral to the overall mass balance is negligible. Based on this observation Equation 5.50 can be simplified mid the appropriate equation for a conservation of mass in this process can be obtained (i.e., V Ur) = 0). Using this, Equation 5.18 can be simplified and the appropriate species balance equation for the IP process can be obtained. This equation is similar to the equation obtained for the RTM process. 

The species balance equations (13) and (14) can be solved for the steady-state values of u and v, and the resulting expressions substituted in the temperature equation. After using eq. (16) the expression... 

In order to capture the effects of non-uniform reactant feeding at the reactor entrance, we rederive our two-mode models by introducing a delta function source in the species balance equation for the y th species [Eq. (123)]... 

Each equation is solved for the complex of interest and substituted into the species balance equation, along with the values for the chloride and sulfate concentrations. The equation is then solved for the pH that gives 1.923 x 10 moles per liter (1 ppm) for the total soluble chromium ( C r( 111)] ). From inspection of the stability complexes, it is clear that hydroxide complexes Cr(III) much more strongly than does either sulfate or chloride. Accordingly, the pH needed to attain 1 ppm Cr(III) is 8.12, regardless of whether one considers complexation by sulfate and chloride at this pH the dominant soluble complex is t rbOH Had complexation by hydroxide not been considered, the minimum pH would have been erroneously calculated as 5.51. 

The fundamental derivation of the species balance equation for systems with homogeneous chemical reactions, starting out from from a Boltzmaim type of equation, is briefly discussed in the following subsection. 

The species balance equations are normally solved with the standard Danck-werts boundary conditions [26]. 

Species Balance Equation and Reactor Design Equation... 

To derive the design equation of a plug-flow reactor with a distributed feed, we write a species balance equation for any species, say species j, that is not injected along the reactor over reactor element dV. Since the species is not fed or withdrawn, its molar balance equation is... 

To derive a design equation, we write a species balance equation for species j that is not removed from the reactor ... 

To derive the design equation of a recycle reactor, we consider a differential reactor element, dV, and write a species balance equation over it for species j ... 

For Ihe case of a perfect gas mixture whose molar concentration is independent of composition, the continuity and species balance equations reduce to the slataments... 

On the basis of Eq. (12), and mechanism (14) [64,68], the species balance equations were solved by eliminating the time variable (phase-plane model). The relevant rate equations are... 

A dilute solution of constam density and bulk composition wj is in contact with a sphere where the mass fraction is h>](. For dissolution or crystal growth would be determined from solubility while for ion exchange it could be nearly zero at low saturations. The steady-state continuity and species balance equations, considering only radial variations, are... 

Correct species, balanced equation, valid molecular interpretation... 

Note that in the formulation of the continuity, the species balance equation for the fine particles, and the deposition/detach-ment rate equation, the fraction of the collector surface area avail-able/not available for fine particle adhesion/detachment (yy) corresponds to the regions where the shear stress acting on the collector is lower than the critical shear stress [131]. The expression for the fraction of the collector surface area available/not available for the fine particle adhesion/detachment is [131]... 

The velocities at the inlet are specified in Dirichlet-typeboundary conditions. At the outlet, an open boundary condition is used [124], which implies, except for pressure, zero gradients for all flowvariables normal to the outflow boundary. The liquid holdup at the reactor inlet is evaluated assuming dugjdz = dui/dz = 0 and fully wetted particles and combining the momentum balance equations for the gas and liquid phases. Danckwerts boundary conditions are used for unsteady-state mass balance equations for planktonic cells and x-size aggregates and species balance equations in gas and liquid phases [137]. 

The dotted line represents the concentration of the phenol calculated mathematically using the species balance equations listed earlier. The washing out effect due to dilution is shown and correlates well with the measured bioluminescence wash out pattern. Figure 5 shows the results of TV 1061 induced with ethanol using the step injection method. Again the bioluminescence leveled off, here around a value of 3.5, and follows the wash out pattern as well. These results demonstrate the... 

In statistically steady state, the species balance equation over the small scale structures expresses the species transfer rate between the large and the small scale structures compensating the reaction rate ... 

Both the simulation model for the detailed calculation of the flue gas pattern in the furnace and the process gas pattern in the cracking tubes are based on the Reynolds-Averaged Navier-Stokes mass, momentum, energy, and species balance equations described in Section 12.5. Turbulent momentum, species, and... 

To account to a certain extent for the effects of macro-mixing in the species balance equations, so-called multi-zone models have been developed. The reactor is divided in multiple, not necessarily geometrical, zones and, within each zone, an idealized mixing or flow pattern is assumed in the derivation of the species... 

A multi-zone model similar to the finite volume methods used in Computational Fluid Dynamics (CFD, see Section 12.5) is based on geometrical zones and assumes uniform conditions within each zone, that is, completely mixed (CSTR) zones, while essentially allowing conditions to differ between zones. For statistically stationary flow, the species balance equations for species A in zone i of iVcan then be written as ... 

The underlying kinetic models and reaction mechanisms are described later. In both models, the variable represents a gas-phase concentration, x, a surface concentration, and is a surface capacitance factor. Therefore, the function Fi(x2 X2) accounts for flow terms (in a CSTR species balance equation) as well as chemisorption and perhaps other reaction terms. The function 2 (x], X2), on the other hand, accounts only for surface rate processes. The variable X2 is a "latent" variable, not measurable in situ in the unsteady state presently. 

Consider the unit square domain of the sine flow, where the right side is filled with reactant A and the left side contains only B. The initial interface between the two components is a vertical line along the center and one edge of the box. At t = 0 convective mixing is turned on in the model simultaneously with diffusion and reaction. The species balance equation in a dimensionless form for the flow with reaction, diffusion, and convection is... 

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