Conservation equations analytical solutions

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... 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

Strehlow (1975) achieved a solution by conducting a mass balance over the flow field. Such a balance can be drawn up under the assumptions of similarity and a constant density between shock and flame. The assumption of constant density violates the momentum-conservation equation, and is a drastic simplification. The maximum overpressure is, therefore, substantially underestimated over the entire flame speed range. An additional drawback is that the relationship of overpressure to flame speed is not produced in the form of a tractable analytical expression, but must be found by trial and error. 

Since the ORR is a first-order reaction following Tafel kinetics, the solution of the mass conservation equation (eq 23) in a spherical agglomerate yields an analytic expression for the effectiveness factor... 

The limitations encountered when obtaining an analytical solution to the conservation equations, as in the present work, differ from those encountered applying direct computational methods. For example, the cost of numerical computations is dependent on the grid and, especially, on the number of species for which conservation equations must be solved additional reactions do not add significantly to the computational effort. With RRA techniques, further limitations arise on the number of different reaction paths that can conveniently be included in the analysis. The analysis typically follows a sequence of reactions that make up the main path of oxidation, the most important reactions, while parallel sequences are treated as perturbations to the main solution and often are sufficiently unimportant to be neglected. The first step thus identifies a skeletal mechanism of 63 elementary steps by omitting the least important steps of the detailed mechanism [44]. 

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. 

In reality however, situations also exist where a more complex form of the rate expression has to be applied. Among the numerous possible types of kinetic expressions two important cases will be discussed here in more detail, namely rate laws for reversible reactions and rate laws of the Langmuir-Hinshelwood type. Basically, the purpose of this is to point out additional effects concerning the dependence of the effectiveness factor upon the operating conditions which result from a more complex form of the rate expression. Moreover, without going too much into the details, it is intended at least to demonstrate to what extent the mathematical effort required for an analytical solution of the governing mass and enthalpy conservation equations is increased, and how much a clear presentation of the results is hindered whenever complex kinetic expressions are necessary. 

Mathematically, the combustion process has been modelled for the most general three-dimensional case. It is described by a sum of differential equations accounting for the heat and mass transfer in the reacting system under the assumption of energy and mass conservation laws At present, it is impossible to obtain an analytical solution for the three-dimensional form. Therefore, all the available condensed system combustion theories are based on simplified models with one-dimensional or, at best, two-dimensional heat and mass transfer schemes. In these models, the kinetics of the chemical processes taking place in the phases or at the interface is described by an Arrhenius equation (exponential relationship between the reaction rate constant and temperature), and a corresponding reaction order with respect to reactant concentrations. 

These models are based on ID mass and current conservation equations, coupled with Tafel equations in each electrode. Various versions of such models are used nowadays [154-158]. Simplified versions of these models were even amenable to analytical treatment and a number of important aspects of electrode performance were rationalized on the basis of analytical solutions (cf. the preceding section). These models clarified a number of features of PEFC operation and allowed to identify important sources of voltage losses in the cell. 

While the film and surface-renewal theories are based on a simplified physical model of the flow situation at the interface, the boundary layer methods couple the heat and mass transfer equation directly with the momentum balance. These theories thus result in anal3dical solutions that may be considered more accurate in comparison to the film or surface-renewal models. However, to be able to solve the governing equations analytically, only very idealized flow situations can be considered. Alternatively, more realistic functional forms of the local velocity, species concentration and temperature profiles can be postulated while the functions themselves are specified under certain constraints on integral conservation. Prom these integral relationships models for the shear stress (momentum transfer), the conductive heat flux (heat transfer) and the species diffusive flux (mass transfer) can be obtained. 

Henry [27,28] was who the first started theoretical investigation of this phenomenon. He proposed a system of differential equations to describe the coupled heat and moisture diffusion into bales of cotton. Two of the equations involve the conservation of mass and energy, and the third relates fiber moisture content with the moisture in the adjacent air. Since these equations are non-linear, Henry made a number of simplifying assumptions to derive an analytical solution. 

As discussed previously, the radiative transfer equation is written in terms of radiation intensity, which is a function of seven independent parameters. The RTE is developed phenomenologically and is a mathematical expression of a physical model (i.e., the conservation of the radiative energy). It is a complicated integro-differential equation. There is no available analytical solution to the RTE in its general form. In order to solve it, physical and mathematical approximations are to be introduced individually or in tandem. 

The electrical potential distribution is a function of the electrical resistance of the soil and therefore depends on the instantaneous local concentration and mobility of all ions existing in the pore water of the soil, which ultimately determine the electrical resistance evolution. Therefore, a set of conservation equation, one for each ion, has to be simultaneously integrated, with the difficulty that aU these differential equations are strongly coupled through the electrical potential distribution. Therefore, an analytical solution would be very difficult to obtain, if at all possible. So, a numerical solution must be used instead. 

The conservation equations for mass and momentum are more complex than they appear. They are nonlinear, coupled and difficult to solve. Only in a small number of cases - mostly Mly developed flows with constant viscosity in simple geometries e.g. in channels, pipes, between parallel plates - it is possible to obtain an analytical solution of the Navier-Stokes equations. In this chapter we will consider such a type of elementary flow, to show, how simple geometries and physics have to be for an analytical solution. Further elementary fluid flows can be found in a multitude of books about fluid mechanics. We follow in this chapter the accomplishments of (Sabersky and Acosta 1964). 

In a more recent study. Das and Chakraborty [9] presented analytical solutions for velocity, temperature, and concentration distribution in electroosmotic flows of non-Newtonian fluids in microchannels. A brief description of their transport model is summarized here, for the sake of completeness. A schematic diagram of the parallel plate microchannel configuration, as considered by the above authors, is depicted in Fig. 2. The bottom plate is denoted as y = H and top plate as y = +H. A potential gradient is applied along the axis of the channel, which provides the necessary driving force for electroosmotic flow. The governing equations appropriate to the physical problem are the equations for conservation... 

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