Governing equation

The conservation equations of mass, momentum, and energy of a single-phase flow can be obtained by using the general conservation equation derived previously. 

For the total mass conservation of a single-phase fluid, / represents the fluid density p. jr represents the diffusional flux of total mass, which is zero. For flow systems without chemical reactions, d = 0. Therefore, from Eq. (5.12), we have the continuity equation as 

For the momentum conservation of a single-phase fluid, the momentum per unit volume / is equal to the mass flux pU. The momentum flux is thus expressed by the stress tensor i/r = (pi — t). Here p is the static pressure or equilibrium pressure / is a unit tensor and r is the shear stress tensor. Since 1 = —pf where / is the field force per unit mass, Eq. (5.12) gives rise to the momentum equation as 

pi is the second viscosity or bulk viscosity, which reflects the deviation of the averaged pressure from the pressure at equilibrium due to nonuniformity in the local velocity distribution. 5y is the Kronecker delta, p is given by 

The energy equation expressed in terms of internal energy becomes 

The use of governing equations has the merit of tying specific dimensionless parameters to particular physical phenomena (Glicksman, 1984,1988). If the proper equations can be written, even if they cannot be solved, they yield considerable insight into the process. 

Early derivation of these dimensionless parameters was based on a continuum model (Scharff et al., 1978 Glicksman, 1984). Inclusion of the individual particle approach extends the results to instances where a continuum model may not be applicable. 

Note /3 is, in general, not a constant, rather it must be found from a general expression for the drag force. 

Given the uncertainty in the form of the stress tensors, many authors have adopted a form analogous to single phase Newtonian fluid relating the stress terms to the pressure and viscosity of the fluid and particle phase, respectively. 

For larger particles, the nature of interparticle forces is still unresolved. A typical operating condition for most commercial gas solid beds is well beyond the point of minimum bubbling. It might be expected that 

To derive the governing equations we need to identify each independent chemical reaction that can occur in the system. It is possible to write many more reactions than are independent in a geochemical system. The remaining or 

TABLE 3.1 Constituents of a geochemical model the cast of characters 

A Aqueous species in the basis, the basis species Aj Other aqueous species, the secondary species 

A All minerals, even those that do not exist in the system Am Gases of known fugacity 

Since a geochemical model needs to be cast in general form, the species occurring in reactions are represented symbolically (Table 3.1). Depending on the nature of the problem, we have chosen a basis 

We will derive the governing equations for a buoyant plume with heat added just at its source, approximated here as a point. For a two-dimensional planar plume, this is a line. Either could ideally represent a cigarette tip, electrical resistor, a small fire or the plume far from a big fire where the details of the source are no longer important. We list the following assumptions  

There is an idealized point (or line) source with firepower, Q, losing radiant fraction, X,.  

The plume is Boussinesq, or of constant density px except in the body force term. 

The pressure at any level in the plume is due to an ambient at rest, or hydrostatic . 

Properties are assumed uniform across the plume at any elevation, z. This is called a top-hat profile as compared to the more empirically correct Gaussian profile given in Equation (10.1). 

At this point we can derive a set of governing equations that fully describes the equilibrium state of the geochemical system. To do this we will write the set of independent reactions that can occur among species, minerals, and gases in the system and set forth the mass action equation corresponding to each reaction. Then we will derive a mass balance equation for each chemical component in the system. Substituting the mass action equations into the mass balance equations gives a set 

In addition the following equation holds for the volume V(t) occupied by the material  

The following notation is used in (9.1)-(9.4) and throughout this paper. 

V is the material velocity. a is the stress tensor. g is the acceleration of gravity. e is the internal energy per unit mass. h is the energy flux. 

Note that the spatial velocity (u) is arbitrary and may be the material velocity (u). If the spatial velocity is the material velocity (u = v), then the region of space moves with the material and the Lagrangian forms of the equations are generated. If the spatial velocity is zero, then the region of space is fixed and the equations take the Eulerian form. 

It is worth investigating the time derivatives and demonstrating how to derive (9.1)-(9.4) from the more familiar forms of the conservation equations. The more familiar Lagrangian derivative djdt and d jdt are related by [9] 

Proceeding as in the previous section, the appropriate dimensionless reaction-diffusion equation can be written in terms of the wavefront coordinates as 

THE UNIMOLECULAR DECOMPOSITION FLAME WITH LEWIS NUMBER OF UNITY 

Let us now restrict our attention to deflagrations with Mq 1, retaining all of the assumptions of the preceding section. If cp and d p/d remain small compared with I/Ml (as is, in fact, found to be true from the deflagration solution), then equations (27) and (31) [with F given by equation (32)] imply that 

Since equation (39) represents an approximate solution to equation (35), the only differential equation that remains to be solved is equation (36), in which F (t, (p) = F (t, (p(z)) depends only on t. By utilizing equations (38) and (41) in equation (34), we can obtain the t dependence of explicitly. When this result is substituted into equation (36), we find that 

So that A may be treated as a constant without complicating the definition of CO, we shall (solely for simplicity) employ a final assumption  

Two realistic cases in which the nonessential assumption 11 is valid are 

The propellant is assumed to undergo an irreversible thermal decomposition process in the condensed phase. The simplest description of this process is a single-step, unimolecular reaction with the initial reactant species-A (formula weight W) going to intermediate species-B (same formula weight W) as shown in Fig. 1, 

Based on this single-step mechanism, the one-dimensional, steady conservation of mass (Ah-B), species-A, and energy equations for the condensed phase (x 0) can be written as 

The condensed phase density p, specific heat C, thermal conductivity A c, and radiation absorption coefficient Ka are assumed to be constant. The species-A equation includes only advective transport and depletion of species-A (generation of species-B) by chemical reaction. The species-B balance equation is redundant in this binary system since the total mass equation, m = constant, has been included the mass fraction of B is 1-T. The energy equation includes advective transport, thermal diffusion, chemical reaction, and in-depth absorption of radiation. Species diffusion d Y/cbfl term) and mass/energy transport by turbulence or multi-phase advection (bubbling) which might potentially be important in a sufficiently thick liquid layer are neglected. The radiant flux term qr 

By introducing a reference mass flux nir and the following non-dimensional parameters and variables, f = T/(Tj- -Tg), x-xm Clkg, Ag = PgAghglCnir, 

The last equation, which is an energy balance on the condensed phase, can be written as 

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Equations (1.6) and (1.7) are used to formulate explicit relationships between the extra stress components and the velocity gradients. Using these relationships the extra stress, t, can be eliminated from the governing equations. This is the basis for the derivation of the well-known Navier-Stokes equations which represent the Newtonian flow (Aris, 1989). 

Components of the governing equations of the process can be decoupled to develop a solution scheme for a three-dimensional problem by combining one- and two-dimensional analyses. 

The standard Galerkin technique provides a flexible and powerful method for the solution of problems in areas such as solid mechanics and heat conduction where the model equations arc of elliptic or parabolic type. It can also be used to develop robust schemes for the solution of the governing equations of... 

U-V-P schemes belong to the general category of mixed finite element techniques (Zienkiewicz and Taylor, 1994). In these techniques both velocity and pressure in the governing equations of incompressible flow are regarded as primitive variables and are discretized as unknowns. The method is named after its most commonly used two-dimensional Cartesian version in which U, V and P represent velocity components and pressure, respectively. To describe this scheme we consider the governing equations of incompressible non-Newtonian flow (Equations (1.1) and (1.4), Chapter 1) expressed as... 

In the continuous penalty technique prior to the discretization of the governing equations, the pressure in the equation of motion is substituted from Fquation (3.6) to obtain... 

For simplicity, we define T - and T (A iooTe/At). As explained by Luo and Tanner (1989), the decoupled method requires a suitable variable transfonna-tion in the governing equations (3.20) and (3.21). This is to ensure that the discrete momentum equations always contain the real viscous term required to recover the Newtonian velocity-pressure formulation when Ws approaches zero. This is achieved by decomposing the extra stress T as... 

Governing equations in two-dimensional Cartesian coordinate systems... 

The described continuous penaltyf) time-stepping scheme may yield unstable results in some problems. Therefore we consider an alternative scheme which provides better numerical stability under a wide range of conditions. This scheme is based on the U-V-P method for the slightly compressible continuity equation, described in Chapter 3, Section 1.2, in conjunction with the Taylor-Galerkin time-stepping (see Chapter 2, Section 2.5). The governing equations used in this scheme are as follows... 

To develop the scheme we start with the normalization of the governing equations by letting... 

The selection of a time increment dependent on parameter a (i.e. carrying out Taylor series expansion at a level between successive time steps of n and n+Y) enhances the flexibility of the temporal discretizations by allowing the introduction of various amounts of smoothing in different problems. The first-order time derivatives are found from the governing equations as... 

The governing equations used in this case are identical to Equations (4.1) and (4.4) describing the creeping flow of an incompressible generalized Newtonian fluid. In the air-filled sections if the pressure exceeds a given threshold the equations should be switched to the following set describing a compressible flow... 

In order to account for the heat loss through the metallic body of the cone, a heat conduction equation, obtained by the elimination of the convection and source terms in Equation (5.25), should also be incorporated in the governing equations. 

A similar approximation should be applied to the components of the equation of motion and the significant terms (with respect to ) consistent with the expanded constitutive equation identified. This analy.sis shows that only FI and A appear in the zero-order terms and hence should be evaluated up to the second order. Furthermore, all of the remaining terms in Equation (5.29), except for S, appear only in second-order terms of the approximate equations of motion and only their leading zero-order terms need to be evaluated to preserve the consistency of the governing equations. The term E, which only appears in the higlier-order terms of the expanded equations of motion, can be evaluated approximately using only the viscous terms. Therefore the final set of the extra stress components used in conjunction with the components of the equation of motion are... 

We start with the governing equations of the Stokes flow of incompressible Newtonian fluids. Using an axisymraetric (r, z) coordinate system the components of the equation of motion are hence obtained by substituting the shear-dependent viscosity in Equations (4.11) with a constant viscosity p, as... 

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