Balance species

From the above species constraints (Equations 17.4i to 17.4iii), we also notice that we have four unknown variables, and that the constraints provide us with only three equations we therefore have one degree of freedom in our process. This allows us to evaluate various options for the process. From the above equality constraints (Equations 17.4i to 17.4iii), we also note that the amount of water is fixed simply by the species balance, and that these species (constraints) relationships are linear. 

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... 

CFD may be loosely thought of as computational methods applied to the study of quantities that flow. This would include both methods that solve differential equations and finite automata methods that simulate the motion of fluid particles. We shall include both of these in our discussions of the applications of CFD to packed-tube simulation in Sections III and IV. For our purposes in the present section, we consider CFD to imply the numerical solution of the Navier-Stokes momentum equations and the energy and species balances. The differential forms of these balances are solved over a large number of control volumes. These small control volumes when properly combined form the entire flow geometry. The size and number of control volumes (mesh density) are user determined and together with the chosen discretization will influence the accuracy of the solutions. After boundary conditions have been implemented, the flow and energy balances are solved numerically an iteration process decreases the error in the solution until a satisfactory result has been reached. 

Definitions for the variables and constants appearing in eqns. 1 and 2 are given in the nomenclature section at the end of this paper. The first of these equations represents a mass balance around the reactor, assuming that it operates in a differential manner. The second equation is a species balance written for the catalyst surface. The rate of elementary reaction j is represented by rj, and v j is the stoichiometric coefficient for component i in reaction j. The relationship of rj to the reactant partial pressures and surface species coverages are given by expressions of the form... 

An overall and a component species balance can be performed to represent the mixing process. Because a reaction does not occur during mixing, moles are conserved and it follows that... 

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 ... 

Figure 8.33 shows in detail the effect of the single rate constants on the forward velocity at the various pH-Pco2 conditions T =25 °C). Although the individual reactions 8.290.1-8.290.3 take place simultaneously over the entire compositional field, the bulk forward rate is dominated by reactions with single species in the field shown away from steady state, reaction 8.290.1 is dominant, within the stippled area the effects of all three individual reactions concur to define the overall kinetic behavior, and along the lines labeled 1, 2, and 3 the forward rate corresponding to one species balances the other two. 

Since we can also write the species balances in terms of a single reactant species A, we actually usually solve the equations... 

These configurations are essentially two reactors connected by the heat exchange area A, and we will consider this in more detail in a later chapter in connection with multiphase reactors. We do not need to write a species balance on the coolant, but the energy balance in the jacket or cooling coil is exactly the same as for the reactor except that we omit the reaction generation term. 

For a multicomponent fluid (the only situation of interest with chemical reactions) we next have to solve mass balances for the individual chemical species. This has been implicitly the subject of this book until now. The species balance is written as flow in minus flow... 

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. 

Chemical Species Balances Decay Factors for Reactive Species The CEB method can be extended to families of chemical compounds, such as polycyclic aromatic hydrocarbons (PAH), while taking chemical reaction into account. Formally this can be done by writing chemical species balances in a somewhat different fashion from the CEB formulation ... 

The chemical species balance method can be extended to first-order chemical decay processes as follows ... 

Duval, M.M., (1980) "Source Resolution Studies of Ambient Polycyclic Aromatic Hydrocarbons in the Los Angeles Atmosphere Application of a Chemical Species Balance Method with First Order Chemical Decay", Thesis, Master of Science in Engineering, UCLA. 

Species balances written for each of the components lead to the following set of differential equations ... 

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 above approach of integrating analytically (under certain assumptions) across the porous wall the species balances to obtain local soot consumption rates can be extended for the case of more reactions occurring in the porous wall. In the presence of a precious metal catalyst, the hydrocarbons and the carbon monoxide of the exhaust gases are also oxidized. It can be assumed that all the reactions in the porous wall occur hierarchically (according to their... 

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... 

Rate Expressions. A major difficulty in CVD reactor modeling is the choice of appropriate rate expressions, Rp for the gas-phase-species balance and the surface boundary conditions. As described previously (see Nucleation and Growth Modes), most of CVD chemical kinetics is unknown. Therefore, rate parameters may have to be estimated from experimental growth data as part of the reactor-modeling effort. 

Finally, a steady 1-D lumped stack model is introduced which uses a 0-D lumped approach for each cell in the stack. The model takes the current and power produced by each cell in the stack as input and predicts the 1 -D temperature distribution across the cells of the stack. Such models have the advantage of faster calculation time and are thus better suited for initial design calculations and control system modeling. In this model, each fuel cell is divided into three components, air channel, fuel channel and solid region (electrodes, electrolyte and the interconnect). The control volumes used for air and fuel channel components are shown by the dashed lines in Figure 5.6. The specie concentrations at the exit of air and fuel channels could be calculated using the mass and specie balances for these control volumes which are in the form... 

The species balance relation Eq. 13.2-8 is transformed to a difference equation using the forward difference on the time derivative and the backward difference on the space derivative. The finite difference form of the x-momentum equation (Eq. 13.2-25) is obtained by using the forward difference on all derivatives, and is solved by the Crank-Nicolson method. The same is true for the energy equation (Eq. 13.2-26). 

An unsteady state species balance on water yields —----= -yi-Vdot... 

Substituting the species balances into the total balance yields... 

The material balances for the reactant and product species Balance for species A... 

The numerical constant 6 in the preceding equation implies the cubic domain assumed by de Gennes and Taupin,7 and the surface area aF(gsi)avg is that of a cube of characteristic size k- For other domain geometries, such as the polyhedra considered by Talmon and Prager,6 this numerical coefficient will be different. The relative proportions of the oil and water domains are related to the persistence length k and the volume fraction d s of the surfactant present through the species balance relation... 

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)]... 

On the other hand, the acidity in acid-exchanged synthetic mordenite has been shown by Flank and Skeels in 1977 to arise from either H30+ or, following removal of adsorbed water at elevated temperatures, H+ species balancing framework negative charges (3). The same study showed that the acidity of calcined NH4+-mordenite arises from two separate and distinct acid centers. Nearly two-thirds of the acidity is due to the presence of H30+ or H+ species. The remaining third of the acidity is due to the formation of hydroxoaluminum cations during the thermal treatment. 

The species balance in a plug flow (Fig. 3.25) is carried out in an elementary dx length of the control volume the result is the partial differential equation (3.73) where w is the velocity of the fluid moving with a plug flow pattern. Then, the relation between the flow rate and the section crossed by flow becomes ... 

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