Conservation equations boundary conditions

To solve the conservation equations, boundary conditions are needed. For the momentum equation, the flow velocity at wall is fixed. It is generally assumed that the fluid molecules near the wall are in equilibrium with those of the wall and the fluid velocity is written as ... 

In fluid mechanics, the velocity field is solved by using the differential equations for mass and momentum conservation with boundary conditions... 

The mathematical theory is rather complex because it involves subjecting the basic equations of motion to the special boundary conditions of a surface that may possess viscoelasticity. An element of fluid can generally be held to satisfy two kinds of conservation equations. First, by conservation of mass. 

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]. 

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 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. 

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

Heat Transfer to the Containing Wall. Heat transfer between the container wall and the reactor contents enters into the design analysis as a boundary condition on the differential or difference equation describing energy conservation. If the heat flux through the reactor wall is designated as qw, the heat transfer coefficient at the wall is defined as... 

For a given mass transfer problem, the above conservation equations must be complemented with the applicable initial and boundary conditions. The problem of finding the mathematical function that represents the behaviour of the system (defined by the conservation equations and the appropriate set of initial and boundary conditions), is known as a boundary value problem . The boundary conditions specifically depend on the nature of the physicochemical processes in which the considered component is involved. Various classes of boundary conditions, resulting from various types of interfacial processes, will appear in the remainder of this chapter and Chapters 4 and 10. Here, we will discuss some simple boundary conditions using examples of the diffusion of a certain species taken up by an organism ... 

Thermal/structural response models are related to field models in that they numerically solve the conservation of energy equation, though only in solid elements. Finite difference and finite element schemes are most often employed. A solid region is divided into elements in much the same way that the field models divide a compartment into regions. Several types of surface boundary conditions are available adiabatic, convection/radiation, constant flux, or constant temperature. Many ofthese models allow for temperature and spatially dependent material properties. 

The numerical jet model [9-11] is based on the numerical solution of the time-dependent, compressible flow conservation equations for total mass, energy, momentum, and chemical species number densities, with appropriate in-flow/outfiow open-boundary conditions and an ideal gas equation of state. In the reactive simulations, multispecies temperature-dependent diffusion and thermal conduction processes [11, 12] are calculated explicitly using central difference approximations and coupled to chemical kinetics and convection using timestep-splitting techniques [13]. Global models for hydrogen [14] and propane chemistry [15] have been used in the 3D, time-dependent reactive jet simulations. Extensive comparisons with laboratory experiments have been reported for non-reactive jets [9, 16] validation of the reactive/diffusive models is discussed in [14]. 

In practice, of course, it is rare that the catalytic reactor employed for a particular process operates isothermally. More often than not, heat is generated by exothermic reactions (or absorbed by endothermic reactions) within the reactor. Consequently, it is necessary to consider what effect non-isothermal conditions have on catalytic selectivity. The influence which the simultaneous transfer of heat and mass has on the selectivity of catalytic reactions can be assessed from a mathematical model in which diffusion and chemical reactions of each component within the porous catalyst are represented by differential equations and in which heat released or absorbed by reaction is described by a heat balance equation. The boundary conditions ascribed to the problem depend on whether interparticle heat and mass transfer are considered important. To illustrate how the model is constructed, the case of two concurrent first-order reactions is considered. As pointed out in the last section, if conditions were isothermal, selectivity would not be affected by any change in diffusivity within the catalyst pellet. However, non-isothermal conditions do affect selectivity even when both competing reactions are of the same kinetic order. The conservation equations for each component are described by... 

Because both kinetic constants k i and k 2 are exponentially dependent on the temperature, Tg, within the pellets (according to an Arrhenius form of temperature dependence) and the reactant concentration,, appears explicitly in the three mass conservation equations and also the heat balance equation, the problem must be solved numerically, rather than analytically. The boundary conditions at the pellet centre are... 

The existence of radial temperature and concentration gradients means that, in principle, one should include terms to account for these effects in the mass and heat conservation equations describing the reactor. Furthermore, there should be a distinction between the porous catalyst particles (within which reaction occurs) and the bulk gas phase. Thus, conservation equations should be written for the catalyst particles as well as for the bulk gas phase and coupled by boundary condition statements to the effect that the mass and heat fluxes at the periphery of particles are balanced by mass and heat transfer between catalyst particles and bulk gas phase. A full account of these principles may be found in a number of texts [23, 35, 36]. 

The physical significance of these boundary conditions is as follows. Equation (26a) represents the fact that just before the tracer is injected into the system the concentration is everywhere zero. Equations (26b) and (26g) are obvious since a finite amount of tracer is injected. Equations (26d) and (26e) follow from conservation of mass at the boundaries between the sections (W4) the total mass flux entering the boundary must equal that leaving. Equations (26c) and (26f) are based on the physically intuitive argument that concentration should be continuous in the neighborhood of any point. These boundary conditions will be used extensively in the subsequent derivations. 

The fundamental physical laws governing motion of and transfer to particles immersed in fluids are Newton s second law, the principle of conservation of mass, and the first law of thermodynamics. Application of these laws to an infinitesimal element of material or to an infinitesimal control volume leads to the Navier-Stokes, continuity, and energy equations. Exact analytical solutions to these equations have been derived only under restricted conditions. More usually, it is necessary to solve the equations numerically or to resort to approximate techniques where certain terms are omitted or modified in favor of those which are known to be more important. In other cases, the governing equations can do no more than suggest relevant dimensionless groups with which to correlate experimental data. Boundary conditions must also be specified carefully to solve the equations and these conditions are discussed below together with the equations themselves. 

A more sophisticated approach is to avoid the postulate of a shock and instead to state the differential equations of conservation of mass, momentum, and energy to include more properties of a real fluid. Including the effects of viscosity, heat conditions, and diffusion along with chem reaction gives eqs with a unique solution for given boundary conditions and so solves the determinacy problem. The boundary conditions are restricted by the assumption that the reaction begins and is completed with the region considered. 

In practice the equation of motion is solved first without considering the constraint force and in the next step the constraint forces are obtained by correcting the positions such that the molecule conserves its minimum structure, i.e., such that the constraints are fiilfilled. For small molecules direct inversion is possible, for large molecules iterative procedures are applied (60). This means that each constraint is corrected after the other until a certain convergence is reached. This algorithm is called Shake (65). Another important aspect of simulations concerns periodic boundary conditions. A virtual replication of the central box at each of its planes is carried out in order to avoid surface effects. A detailed description can be found in the excellent textbooks of Allen and Tildesley (60) and Frenkel and Smit (61). 

Deriving the compressible, transient form of the stagnation-flow equations follows a procudeure that is largely analogous to the steady-state or the constant-pressure situation. Beginning with the full axisymmetric conservation equations, it is conjectured that the solutions are functions of time t and the axial coordinate z in the following form axial velocity u = u(t, z), scaled radial velocity V(t, z) = v/r, temperature T = T(t, z), and mass fractions y = Yk(t,z). Boundary condition, which are applied at extremeties of the z domain, are radially independent. After some manipulation of the momentum equations, it can be shown that... 

The reaction scheme at and near the phase boundary during the phase transformation is depicted in Figure 10-14. The width of the defect relaxation zone around the moving boundary is AifR, it designates the region in which the relaxation processes take place. The boundary moves with velocity ub(f) and establishes the boundary conditions for diffusion in the adjacent phases a and p. The conservation of mass couples the various processes. This is shown schematically in Figure 10-14b where the thermodynamic conditions illustrated in Figure 10-12 are also taken into account. The transport equations (Fick s second laws) have to be solved in both the a and p... 

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