The Conservation Equations

The component material balance equations for the vapor phase may be written as follows  

The last term in the left-hand side of Eqs. 14.1.1 and 14.1.2 represents the net loss or 

Equation 14.1.5 may also be derived by constructing a material balance around the entire interface. 

The total material balances for the two phases are obtained by summing Eqs. 14.1.1 and 

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These expressions are inserted in the conservation equations, and the boundary conditions provide a set of relationships defining the U and V coefficients [125-129]. 

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

Averaging the velocity using equation 50 yields the weU-known Hagen-Poiseuille equation (see eq. 32) for laminar flow of Newtonian fluids in tubes. The momentum balance can also be used to describe the pressure changes at a sudden expansion in turbulent flow (Fig. 21b). The control surface 2 is taken to be sufficiently far downstream that the flow is uniform but sufficiently close to surface 3 that wall shear is negligible. The additional important assumption is made that the pressure is uniform on surface 3. The conservation equations are then applied as follows ... 

By applying the conservation equations of mass and energy and by neglecting the small pressure changes across the flame, the thickness of the preheating and reaction 2ones can be calculated for a one-dimensional flame (1). 

The physics and modeling of turbulent flows are affected by combustion through the production of density variations, buoyancy effects, dilation due to heat release, molecular transport, and instabiUty (1,2,3,5,8). Consequently, the conservation equations need to be modified to take these effects into account. This modification is achieved by the use of statistical quantities in the conservation equations. For example, because of the variations and fluctuations in the density that occur in turbulent combustion flows, density weighted mean values, or Favre mean values, are used for velocity components, mass fractions, enthalpy, and temperature. The turbulent diffusion flame can also be treated in terms of a probabiUty distribution function (pdf), the shape of which is assumed to be known a priori (1). 

Entrance effects are usually not significant industrially if L/D > 60. Below this limit Nusselt recommended the conservative equation for 10 < L/D < 400 and properties evaluated at bulk temperature... 

Examples Four examples follow, illustrating the apphcation of the conservation equations to obtain useful information about fluid flows. 

Jump conditions Expressions for conservation of mass, momentum, and energy across a steady wave or shock discontinuity ((2. l)-(2.3)). Also known as the conservation equations or the Rankine-Hugoniot relations. 

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]... 

The conservation equations are more commonly written in the initial reference frame (Lagrangian forms). The time derivative normally used is d /dt. Equation (9.5) is used to derive (9.2) from the Lagrangian form of the conservation of mass... 

There are few analytic solutions to the governing equations for interesting problems. The conservation equations are typically solved approximately on digital computers. It is assumed that the sound speeds are real and the system... 

Lagranglan codes are characterized by moving the mesh with the material motion, u = y, in (9.1)-(9.4), [24]. The convection terms drop out of (9.1)-(9.4) simplifying all the equations. The convection terms are the first terms on the right-hand side of the conservation equations that give rise to fluxes between the elements. Equations (9.1)-(9.2) are satisfied automatically, since the computational mesh moves with the material and, hence, no volume or mass flux occurs across element boundaries. Momentum and energy still flow through the mesh and, therefore, (9.3)-(9.4) must be solved. 

The conservation equations developed by Ericksen [37] for nematic liquid crystals (of mass, linear momentum, and angular momentum, respectively) are ... 

Prediction and analysis of crystallizer performance is achieved by constructing models based on conservation equations and rate expressions respectively, as follows. In general form, the conservation equation is given by... 

The Chapman-Jongnet (CJ) theory is a one-dimensional model that treats the detonation shock wave as a discontinnity with infinite reaction rate. The conservation equations for mass, momentum, and energy across the one-dimensional wave gives a unique solution for the detonation velocity (CJ velocity) and the state of combustion products immediately behind the detonation wave. Based on the CJ theory it is possible to calculate detonation velocity, detonation pressure, etc. if the gas mixtnre composition is known. The CJ theory does not require any information about the chemical reaction rate (i.e., chemical kinetics). 

Analytical methods relate the gas dynamics of the expansion flow field to an energy addition that is fully prescribed. A first step in this approach is to examine spherical geometry as the simplest in which a gas explosion manifests itself. The gas dynamics of a spherical flow field is described by the conservation equations for mass, momentum, and energy ... 

The flow field in front of an expanding piston is characterized by a leading gas-dynamic discontinuity, namely, a shock followed by a monotonic increase in gas-dynamic variables toward the piston. If both shock and piston are regarded as boundary conditions, the intermediate flow field may be treated as isentropic. Therefore, the gas dynamics can be described by only two dependent variables. Moreover, the assumption of similarity reduces the number of independent variables to one, which makes it possible to recast the conservation equations for mass and momentum into a set of two simultaneous ordinary differential equations ... 

The major mechanism of a vapor cloud explosion, the feedback in the interaction of combustion, flow, and turbulence, can be readily found in this mathematical model. The combustion rate, which is primarily determined by the turbulence properties, is a source term in the conservation equation for the fuel-mass fraction. The attendant energy release results in a distribution of internal energy which is described by the equation for conservation of energy. This internal energy distribution is translated into a pressure field which drives the flow field through momentum equations. The flow field acts as source term in the turbulence model, which results in a turbulent-flow structure. Finally, the turbulence properties, together with the composition, determine the rate of combustion. This completes the circle, the feedback in the process of turbulent, premixed combustion in gas explosions. The set of equations has been solved with various numerical methods e.g., SIMPLE (Patankar 1980) SOLA-ICE (Cloutman et al. 1976). 

The conservation equation for channel species is [R total] = [AR open ] -T [ARciosecj] + [R... 

The conservation equation for G-protein is [Gtot] = [G] + [RaG], The amount of receptor-activated G-protein expressed as a fraction of total G-protein ([RaG]/[Gtot]) is ... 

We will not list Up or V, the steady detonation particle velocity and detonation product specific volume, as they are completely determined by the conservation equations, namely p /Ro0 md... 

Simultaneous solution of these equilibrium relations (coupled with the conservation equations x+ x-f = 1 and x/ + x/ = 1) gives the coexistence curve for the two-phase system as a function of pressure. 

Consider a steady-state process represented by Figure 3.1. The conservation equation can be written to include the various forms of energy. 

Because the enzyme serves as a catalyst and is not consumed, the conservation equation on the enzyme yields... 

The mathematical model describing the two-phase dynamic system consists of modeling of the flow and description of its boundary conditions. The description of the flow is based on the conservation equations as well as constitutive laws. The latter define the properties of the system with a certain degree of idealization, simplification, or empiricism, such as equation of state, steam table, friction, and heat transfer correlations (see Sec. 3.4). A typical set of six conservation equations is discussed by Boure (1975), together with the number and nature of the necessary constitutive laws. With only a few general assumptions, these equations can be written, for a one-dimensional (z) flow of constant cross section, without injection or suction at the wall, as follows. 

The flux vector accounts for mass transport by both convection (i.e., blood flow, interstitial fluid flow) and conduction (i.e., molecular diffusion), whereas S describes membrane transport between adjacent compartments and irreversible elimination processes. For the three-subcompartment organ model presented in Figure 2, with concentration both space- and time-dependent, the conservation equations are... 

Solution of the conservation equations requires boundary conditions that are dictated by the specifics of the experiment and assumptions of the model. 

We shall call this formulation C. Notice that in this formulation the conservation equation around each vertex is expressed in terms of the pressures at the adjacent vertices, VA and VB. The structure of these equations is related to that of the underlying graph in an interesting manner. 

If the conservation equation for total enzyme concentration (7.3.25) is employed, the last equation becomes... 

F or nonconstant diffusivity, a numerical solution of the conservation equations is generally required. In molecular sieve zeolites, when equilibrium is described by the Langmuir isotherm, the concentration dependence of the intracrystalline diffusivity can often be approximated by Eq. (16-72). The relevant rate equation is ... 

For a linear isotherm (n, = K,c,), this equation is identical to the conservation equation for solid diffusion, except that the solid diffusivity Dsi is replaced by the equivalent diffusivity Dei = EL,/(Ep + ppK,). Thus, Eqs. (16-96) and (16-99) can be used for pore diffusion control with infinite and finite fluid volumes simply by replacing Ds, with Dd. 

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