Species conservation equation derivation

The SGS turbulence model employed is the compressible form of the dynamic Smagorinsky model [17, 18]. The SGS combustion model involves a direct closure of the filtered reaction rate using the scale-similarity filtered reaction rate model. Derivation of the model starts with the reaction rate for the ith species, to i", which represents the volumetric rate of formation or consumption of a species due to chemical reaction and appears as a source term on the right hand side of the species conservation equations ... 

To solve the preceding set of equations, Equation 5.62 is plugged into Equation 5.60. By separately determining the compaction properties of the fiber bed [32] an evolution equation for the pressure can be obtained. Because this is a moving boundary problem the derivative in the thickness direction can be rewritten [32] in terms of an instantaneous thickness. The pressure field can then be solved for by finite difference or finite element techniques. Once the pressure is obtained and the velocity computed, the energy and cured species conservation equations can be solved using the methodology outlined in Section 5.4.1. 

Deriving the species-conservation equation begins with the conservation law for a flowing system... 

In view of equation (39), the similarity in the forms of equation (40) (for the thermal enthalpy Jto p dT) and equation (41) (for the mass fractions 1 ) is striking. Equations (40) and (41) are the energy- and species-conservation equations of Shvab and Zel dovich. The derivation given for these equations required neither that any transport coefficient or the specific heat of the mixture is constant nor that the specific heats of all species are equal. Coupling functions may now be identified from equations (40) and (41). 

In the operator L, the first term represents convection and the second diffusion. Equation (44) therefore describes a balance of convective, diffusive, and reactive effects. Such balances are very common in combustion and often are employed as points of departure in theories that do not begin with derivations of conservation equations. If the steady-flow approximation is relaxed, then an additional term, d(p(x)/dt, appears in L this term represents accumulation of thermal energy or chemical species. For species conservation, equations (48) and (49) may be derived with this generalized definition of L, in the absence of the assumptions of low-speed flow and of a Lewis... 

In deriving the species conservation equations, will be set equal to unity, which is not a summational invariant, since the number of molecules need not be conserved in chemical reactions. With ij/- = 1, equation (31) reduces to... 

In Section 4.2.4, the governing equations of fluid mechanics for a turbulent flow are derived. Similarly, the governing equations for heat transfer and mass transfer can be derived from the principles of energy and mass conservation. In fact, the species conservation equation is an extension of the overall mass conservation (or the continuity) equation. For species i, it has the following form ... 

The following mass transfer (species conservation) equation can be derived from lattice Boltzmann equation after Chapman-Enskog expansion [18] (also see Appendix 3). 

Assume now that the movement of solute species 1 in the solvent does not affect the movement of solute species 2 and vice versa. Following the derivation of the species balance equation (3.2.2) for i = 1, we may obtain a similar species conservation equation for i = 2 ... 

Equations for the rates of change of individual isotopes in a reservoir are not essentially different from the equations for the rates of change of chemical species. Isotopic abundances, however, are generally expressed as ratios of one isotope to another and, moreover, not just as the ratio but also as the departure of the ratio from a standard. This circumstance introduces some algebra into the derivation of an isotopic conservation equation. It is convenient to pursue this algebra just once, as I shall in this section, after which all isotope simulations can be formulated in the same way. I shall use the carbon isotopes to illustrate this derivation, but the... 

This matrix can be derived from Table 6.1 by expressing the unknown species H20 as a function of OH- and H+, then removing the OH" conservation equation, i.e., removing the corresponding column. 

V = s[EP] and the rate law can be derived by using the conservation equation (now containing EP as an additional species), along with two linear differential equations to solve for the three different enzyme forms ... 

The primary purpose of Chapters 2 and 3 is to derive the conservation equations. The conservations equations are partial differential equations where the independent variables are the spatial coordinates and time. Dependent variables are the velocity, pressure, energy, and species composition fields. Inasmuch as we devote some hundred pages to the derivations, it is helpful at this point to have a roadmap for the process. 

The continuity equation is a statement of mass conservation. As presented in Section 3.1, however, no distinction is made as to the chemical identity of individual species in the flow. Mass of any sort flowing into or out of a differential element contributes to the net rate of change of mass in the element. Thus the overall continuity equation does not need to explicitly demonstrate the fact that the flow may be composed of different chemical constituents. Of course, the equation of state that relates the mass density to other state variables does indirectly bring the chemical composition of the flow into the continuity equation. Also, as presented, the continuity-equation derivation does not include diffusive flux of mass across the differential element s surfaces. Moreover there is no provision for mass to be created or destroyed within the differential element s volume. 

In addition to overall mass conservation, we are concerned with the conservation laws for individual chemical species. Beginning in a way analogous to the approach for the overall mass-conservation equation, we seek an equation for the rate of change of the mass of species k, mk. Here the extensive variable is N = mu and the intensive variable is the mass fraction, T = mk/m. Homogeneous chemical reaction can produce species within the system, and species can be transported into the system by molecular diffusion. There is convective transport as well, but it represented on the left-hand side through the substantial derivative. Thus, in the Eulerian framework, using the relationship between the system and the control volume yields... 

In cylindrical coordinates, after we expand out the substantial derivative, the species mass conservation equation becomes... 

Certain numerical methods benefit from writing the convective terms in a conservative form. For example, in the species conservation, show how the continuity equation can be used to write the substantial derivative as... 

The conservation equations for continuous flow of species K will be derived by using the idea of a control volume r t) enclosed by its control surface o t) and lying wholly within a region occupied by the continuum here t denotes the time. In this appendix only, the notation of Cartesian tensors will be used. Let i = 1, 2, 3) denote the Cartesian coordinates of a point in space. In Cartesian tensor notation, the divergence theorem for any scalar function belonging to the Kth continuum a (x, t), becomes... 

This equation derives from the conservation of mass fluxes at the diffusion layer/bulk solution interface [compare Eqs. (155) and (156) in Chapter 1 and Fig. 1]. In Eq. (1), dC/dt is the overall rate of the concentration variations considered, whereas (dC/dt)chem is that arising from possible consumption or production of the species within the bulk solution. The electrochemical rate of production or consumption is given by the first term on the... 

Takeuchi et al. 7 reported a membrane reactor as a reaction system that provides higher productivity and lower separation cost in chemical reaction processes. In this paper, packed bed catalytic membrane reactor with palladium membrane for SMR reaction has been discussed. The numerical model consists of a full set of partial differential equations derived from conservation of mass, momentum, heat, and chemical species, respectively, with chemical kinetics and appropriate boundary conditions for the problem. The solution of this system was obtained by computational fluid dynamics (CFD). To perform CFD calculations, a commercial solver FLUENT has been used, and the selective permeation through the membrane has been modeled by user-defined functions. The CFD simulation results exhibited the flow distribution in the reactor by inserting a membrane protection tube, in addition to the temperature and concentration distribution in the axial and radial directions in the reactor, as reported in the membrane reactor numerical simulation. On the basis of the simulation results, effects of the flow distribution, concentration polarization, and mass transfer in the packed bed have been evaluated to design a membrane reactor system. 

With increasing complexity of the models, one gets more accurate predictions but also the computation times become considerably larger. Keeping in mind that many of these evaporation models have been developed for use in CFD spray simulations, where hundreds of thousands of droplets have to be considered, computational costs become a primary issue. Therefore, the models discussed in this chapter are limited to the second and third category. The presentation starts with the general conservation equations for mass, species and energy, from which the simplified models are derived. 

Derivation of continuity equation, momentum equations, energy equation, and continuity equations of species in this chapter is based on literature [1,2]. Detailed explanation of conservation equations is beyond the scope of this chapter but further information can be found in literature [1-4]. 

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