Differential equation, momentum density

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. 

It is important to determine the partial-differential-equation order. One of the most important reasons to understand order relates to consistent boundary-condition assignment. All the equations are first order in time. The spatial behavior can be a bit trickier. The continuity equation is first order in the velocity and density. The momentum equations are second order on the velocity and first order in the pressure. The species continuity equations are essentially second order in the composition (mass fraction Yy), since (see Eq. 3.128)... 

Ciano et al. (2006) have used a finite element approach to model a tubular cell 0.3 m long. The equations are available in Ciano et al. (2006). Table 7.2 shows the partial differential equations and the mesh characteristics. This model is computationally demanding and the equations have been solved by adopting an iterative procedure. Initial guess values for temperature and current density are assumed (current density is calculated by means of a lumped model, as the function of the average temperature and the cell voltage). Momentum equation and continuity equation are... 

There are basically two ways of modeling a flow field the fluid is either treated as a collection of molecules or is considered to be continuous and indeflnitely divisible - continuum modeling. The former approach can be of deterministic or probabilistic modeling, while in the latter approach the velocity, density, pressure, etc. are aU deflned at every point in space and time, and the conservation of mass, momentum and energy lead to a set of nonlinear partial differential equations (Navier-Stokes). Fluid modeling classiflcation is depicted schematically in Fig. 1. 

An example where the smooth probability density in Euclidean space is not quite the right one is in the setting of conservative (Hamiltonian) systems such as our N-body molecular system, since the evolution is restricted by invariants. The most obvious of these is the energy which we know to be a constant of motion. Therefore we need to work not on open subsets of the phase space R" of our differential equations, but on lower dimensional submanifolds embedded within the phase space, e.g. the energy surface. It will be necessary to assume a density that is defined over the submanifold of constant energy. If other invariants are present, such as fixed total momentum, the discussion would need to be modified to reflect this fact. 

The zone fire models discussed here take the mathematical form of an initial value problem for a system of differential equations. These equations are derived using the conservation of mass or continuily equation, the conservation of energy or the first law of thermodynamics, the ideal gas law, and definitions of density and internal eneigy. The conservation of momentum is ignored. These conservation laws are invoked for each zone or control volume. A zone may consist of a number of interior regions (usually an upper and a lower gas layer), and a number of wall segments. The basic assumption of a zone fire model is that properties such as temperatures can be uniformly approximated throughout the zone. It is remarkable that this assumption seems to hold for as few as two gas layers. 

The analysis of viscoelastic flows includes the solution of a eoupled set of partial differential equations The equations depicting the conservation of mass, momentum, energy, and constitutive equations for a number of pl sical quantities present in the conservation equations such as density, internal energy, heat flux, stress, and so on depend on process [31]. 

Using scaling analysis and perturbation methods, we have been able to derive approximate expressions for the momentum and energy flux in dilute gases and liquids. These methods physically involve formal expansions about local equilibrium states, and the particular asymptotic restrictions have been formally obtained. The flux expressions now involve the dependent transport variables of mass or number density, velocity, and temperature, and they can be utilized to obtain a closed set of transport equations, which can be solved simultaneously for any particular physical system. The problem at this point becomes a purely mathematical problem of solving a set of coupled nonlinear partial differential equations subject to the particular boundary and initial conditions of the problem at hand. (Still not a simple matter see interlude 6.2.)... 

For steady flow of a gas (at a constant mass flow rate) in a uniform pipe, the pressure, temperature, velocity, density, etc. all vary from point to point along the pipe. The governing equations are the conservation of mass (continuity), conservation of energy, and conservation of momentum, all applied to a differential length of the pipe, as follows. 

Here, the superficial velocity, v, represents a fluid state, and the density, p, a fluid property which, for a compressible fluid, can be related to the pressure through an equation of state. The porosity, (p, which is defined as the void fraction within the media, is a macroscopic property of the porous material. Sources and/or sinks located within the physical system are represented using y/. Volume averaging the differential momentum balance for the same physical situation yields Darcy s law ... 

The Navier-Stokes equations are the differential momentum balances for a three-dimensional flow, subject to the assumptions that the flow is laminar and of a constant-density newtonian fluid and that the stress deformation behavior of such a fluid is analogous to the stress deformation behavior of a perfectly elastic isotropic solid. These equations are useful in setting up momentum balances for three-dimensional flows, particularly in cylindrical or spherical geometries. 

The differential momentum balance equation is written for the x and y directions for the control volume dx dy 1). The driving force is the buoyancy force in the gravitational field and is due to the density difference of the fluid. The momentum balance becomes... 

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